# Some Useful Links: Knots in Physics and Number Theory

In modern times, “knots” have been important objects of study in mathematics. These “knots” are akin to the ones we encounter in ordinary life, except that they don’t have loose ends. For a better idea of what I mean, consider the following picture of what is known as a “trefoil knot“:

More technically, a knot is defined as the embedding of a circle in 3-dimensional space. For more details on the theory of knots, the reader is referred to the following Wikipedia pages:

Knot on Wikipedia

Knot Theory on Wikipedia

One of the reasons why knots have become such a major part of modern mathematical research is because of the work of mathematical physicists such as Edward Witten, who has related them to the Feynman path integral in quantum mechanics (see Lagrangians and Hamiltonians).

Witten, who is very famous for his work on string theory (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) and for being the first, and so far only, physicist to win the prestigious Fields medal, himself explains the relationship between knot theory and quantum mechanics in the following article:

Knots and Quantum Theory by Edward Witten

But knots have also appeared in other branches of mathematics. For example, in number theory, the result in etale cohomology known as Artin-Verdier duality states that the integers are similar to a 3-dimensional object in some sense. In particular, because it has a trivial etale fundamental group (which is kind of an algebraic analogue of the fundamental group discussed in Homotopy Theory and Covering Spaces), it is similar to a 3-sphere (recall the common but somewhat confusing notation that the ordinary sphere we encounter in everyday life is called the 2-sphere, while a circle is also called the 1-sphere).

Note: The fact that a closed 3-dimensional space with a trivial fundamental group is a 3-sphere is the content of a very famous conjecture known as the Poincare conjecture, proved by Grigori Perelman in the early 2000’s.  Perelman refused the million-dollar prize that was supposed to be his reward, as well as the Fields medal.

The prime numbers, because their associated finite fields have one cover for every integer, are like circles, and recalling the definition of knots mentioned above, are therefore like knots on this 3-sphere. This analogy, originally developed by David Mumford and Barry Mazur, is better explained in the following post by Lieven le Bruyn on his blog neverendingbooks:

What is the Knot Associated to a Prime on neverendingbooks

Finally, given what we have discussed, could it be that knot theory can “tie together” (pun intended) physics and number theory? This is the motivation behind the new subject called “arithmetic Chern-Simons theory” which is introduced in the following paper by Minhyong Kim:

Arithmetic Chern-Simons Theory I by Minhyong Kim

Of course, it must also be clarified that this is not the only way by which physics and number theory are related. It is merely another way, a new and not yet thoroughly explored one, by which the unity of mathematics manifests itself via its many different branches helping one another.

# Algebraic Spaces and Stacks

We introduced the concept of a moduli space in The Moduli Space of Elliptic Curves, and constructed explicitly the moduli space of elliptic curves, using the methods of complex analysis. In this post, we introduce the concepts of algebraic spaces and stacks, far-reaching generalizations of the concepts of varieties and schemes (see Varieties and Schemes Revisited), that are very useful, among other things, for constructing “moduli stacks“, which are an improvement over the naive notion of moduli space, namely in that one can obtain from it all “families of objects” by pulling back a “universal object”.

We need first the concept of a fibered category (also spelled fibred category). Given a category $\mathcal{C}$, we say that some other category $\mathcal{S}$ is a category over $\mathcal{C}$ if there is a functor $p$ from $\mathcal{S}$ to $\mathcal{C}$ (this should be reminiscent of our discussion in Grothendieck’s Relative Point of View).

If $\mathcal{S}$ is a category over some other category $\mathcal{C}$, we say that it is a fibered category (over $\mathcal{C}$) if for every object $U=p(x)$ and morphism $f: V\rightarrow U$ in $\mathcal{C}$, there is a strongly cartesian morphism $\phi: f^{*}x\rightarrow x$ in $\mathcal{S}$ with $f=p(\phi)$.

This means that any other morphism $\psi: z\rightarrow x$ whose image $p(\psi)$ under the functor $p$ factors as $p(\psi)=p(\phi)\circ h$ must also factor as $\psi=\phi\circ \theta$ under some unique morphism $\theta: z\rightarrow f^{*}x$ whose image under the functor $p$ is $h$. We refer to $f^{*}x$ as the pullback of $x$ along $f$.

Under the functor $p$, the objects of $\mathcal{S}$ which get sent to $U$ in $\mathcal{C}$ and the morphisms of $\mathcal{S}$ which get sent to the identity morphism $i_{U}$ in $\mathcal{C}$ form a subcategory of $\mathcal{S}$ called the fiber over $U$. We will also write it as $\mathcal{S}_{U}$.

An important example of a fibered category is given by an ordinary presheaf on a category $\mathcal{C}$, i.e. a functor $F:\mathcal{C}^{\text{op}}\rightarrow (\text{Set})$; we can consider it as a category fibered in sets $\mathcal{S}_{F}\rightarrow\mathcal{C}$.

A special kind of fibered category that we will need later on is a category fibered in groupoids. A groupoid is simply a category where all morphisms have inverses, and a category fibered in groupoids is a fibered category where all the fibers are groupoids. A set is a special kind of groupoid, since it may be thought of as a category whose only morphisms are the identity morphisms (which are trivially their own inverses). Hence, the example given in the previous paragraph, that of a presheaf, is also an example of a category fibered in groupoids, since it is fibered in sets.

Now that we have the concept of fibered categories, we next want to define prestacks and stacks. Central to the definition of prestacks and stacks is the concept known as descent, so we have to discuss it first. The theory of descent can be thought of as a formalization of the idea of “gluing”.

Let $\mathcal{U}=\{f_{i}:U_{i}\rightarrow U\}$ be a covering (see Sheaves and More Category Theory: The Grothendieck Topos) of the object $U$ of $\mathcal{C}$. An object with descent data is a collection of objects $X_{i}$ in $\mathcal{S}_{U}$ together with transition isomorphisms $\varphi_{ij}:\text{pr}_{0}^{*}X_{i}\simeq\text{pr}_{1}^{*}X_{j}$ in $\mathcal{S}_{U_{i}\times_{U}U_{j}}$, satisfying the cocycle condition

$\displaystyle \text{pr}_{02}^{*}\varphi_{ik}=\text{pr}_{01}^{*}\varphi_{ij}\circ \text{pr}_{12}^{*}\varphi_{jk}:\text{pr}_{0}^{*}X_{i}\rightarrow \text{pr}_{2}^{*}X_{k}$

The morphisms $\text{pr}_{0}:U_{i}\times_{U}U_{j}\rightarrow U_{i}$ and the $\text{pr}_{1}:U_{i}\times_{U}U_{j}\rightarrow U_{j}$ are the projection morphisms. The notations $\text{pr}_{0}^{*}X_{i}$ and $\text{pr}_{1}^{*}X_{j}$ means that we are “pulling back” $X_{i}$ and $X_{j}$ from $\mathcal{S}_{U_{i}}$ and $\mathcal{S}_{U_{j}}$, respectively, to $\mathcal{S}_{U_{i}\times_{U}U_{j}}$.

A morphism between two objects with descent data is a a collection of morphisms $\psi_{i}:X_{i}\rightarrow X'_{i}$ in $\mathcal{S}_{U_{i}}$ such that $\varphi'_{ij}\circ\text{pr}_{0}^{*}\psi_{i}=\text{pr}_{1}^{*}\psi_{j}\circ\varphi_{ij}$. Therefore we obtain a category, the category of objects with descent data, denoted $\mathcal{DD}(\mathcal{U})$.

We can define a functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ by assigning to each object $X$ of $\mathcal{S}_{U}$ the object with descent data given by the pullback $f_{i}^{*}X$ and the canonical isomorphism $\text{pr}_{0}^{*}f_{i}^{*}X\rightarrow\text{pr}_{1}^{*}f_{j}^{*}X$. An object with descent data that is in the essential image of this functor is called effective.

Before we give the definitions of prestacks and stacks, we recall some definitions from category theory:

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is faithful if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is injective for any two objects $x$ and $y$ of $\mathcal{A}$.

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is full if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is surjective for any two objects $x$ and $y$ of $\mathcal{A}$.

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is essentially surjective if any object $y$ of $\mathcal{B}$ is isomorphic to the image $F(x)$ of some object $x$ in $\mathcal{A}$ under $F$.

A functor which is both faithful and full is called fully faithful. If, in addition, it is also essentially surjective, then it is called an equivalence of categories.

Now we give the definitions of prestacks and stacks using the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ we have defined earlier.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is fully faithful, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a prestack.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is an equivalence of categories, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a stack.

Going back to the example of a presheaf as a fibered category, we now look at what it means when it satisfies the conditions for being a prestack, or a stack:

(i) $F$ is a prestack if and only if it is a separated functor,

(ii) $F$ is stack if and only if it is a sheaf.

We now have the abstract idea of a stack in terms of category theory. Next we want to have more specific examples of interest in algebraic geometry, namely, algebraic spaces and algebraic stacks. For this we need first the idea of a representable functor (and the closely related idea of a representable presheaf). The importance of representability is that this will allow us to “transfer” interesting properties of morphisms between schemes such as being surjective, etale, or smooth, to functors between categories or natural transformations between functors. Therefore we will be able to say that a functor or natural transformation is surjective, or etale, or smooth, which is important, because we will define algebraic spaces and stacks as functors and categories, respectively, but we want them to still be closely related, or similar enough, to schemes.

A representable functor is a functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(X,U)$, for some fixed object $U$ of $\mathcal{C}$.

A representable presheaf is a contravariant functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(U,X)$, for some fixed object $U$ of $\mathcal{C}$. If $\mathcal{C}$ is the category of schemes, the latter functor is also called the functor of points of the object $U$.

We take this opportunity to emphasize a very important concept in modern algebraic geometry. The functor of points $h_{U}$ of a scheme $U$ may be identified with $U$ itself. There are many advantages to this point of view (which is also known as functorial algebraic geometry); in particular we will need it later when we give the definition of algebraic spaces and stacks.

We now have the idea of a representable functor. Next we want to have an idea of a representable natural transformation (or representable morphism) of functors. We will need another prerequisite, that of a fiber product of functors.

Let $F,G,H:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors, and let $a:F\rightarrow G$ and $b:H\rightarrow G$ be natural transformations between these functors. Then the fiber product $F\times_{a,G,b}H$ is a functor from $\mathcal{C}^{\text{op}}$ to $\textbf{Sets}$, and is given by the formula

$\displaystyle (F\times_{a,G,b}H)(X)=F(X)\times_{a_{X},G(X),b_{X}}H(X)$

for any object $X$ of $\mathcal{C}$.

Let $F,G:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors. We say that a natural transformation $a:F\rightarrow G$ is representable, or that $F$ is relatively representable over $G$ if for every $U\in\text{Ob}(\mathcal{C})$ and any $\xi\in G(U)$ the functor $h_{U}\times_{G}F$ is representable.

We now let $(\text{Sch}/S)_{\text{fppf}}$ be the site (a category with a Grothendieck topology –  see also More Category Theory: The Grothendieck Topos) whose underlying category is the category of $S$-schemes, and whose coverings are given by families of flat, locally finitely presented morphisms. Any etale covering or Zariski covering is an example of this “fppf covering” (“fppf” stands for fidelement plate de presentation finie, which is French for faithfully flat and finitely presented).

An algebraic space over a scheme $S$ is a presheaf

$\displaystyle F:((\text{Sch}/S)_{\text{fppf}})^{\text{op}}\rightarrow \textbf{Sets}$

with the following properties

(1) The presheaf $F$ is a sheaf.

(2) The diagonal morphism $F\rightarrow F\times F$ is representable.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and a map $h_{U}\rightarrow F$ which is surjective, and etale (This is often written simply as $U\rightarrow F$). The scheme $U$ is also called an atlas.

The diagonal morphism being representable implies that the natural transformation $h_{U}\rightarrow F$ is also representable, and this is what allows us to describe it as surjective and etale, as has been explained earlier.

An algebraic space is a generalization of the notion of a scheme. In fact, a scheme is simply the case where, for the third condition, we have $U$ is the disjoint union of affine schemes $U_{i}$ and where the map $h_{U}\rightarrow F$ is an open immersion. We recall that a scheme may be thought of as being made up of affine schemes “glued together”. This “gluing” is obtained using the Zariski topology. The notion of an algebraic space generalizes this to the etale topology.

Next we want to define algebraic stacks. Unlike algebraic spaces, which we defined as presheaves (functors), we will define algebraic stacks as categories, so we need to once again revisit the notion of representability in terms of categories.

Let $\mathcal{C}$ be a category. A category fibered in groupoids $p:\mathcal{S}\rightarrow\mathcal{C}$ is called representable if there exists an object $X$ of $\mathcal{C}$ and an equivalence $j:\mathcal{S}\rightarrow \mathcal{C}/X$ (The notation $\mathcal{C}/X$ signifies a slice category, whose objects are morphisms $f:U\rightarrow X$ in $\mathcal{C}$, and whose morphisms are morphisms $h:U\rightarrow V$ in $\mathcal{C}$ such that $f=g\circ h$, where $g:U\rightarrow X$).

We give two specific special cases of interest to us (although in this post we will only need the latter):

Let $\mathcal{X}$ be a category fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$. Then $\mathcal{X}$ is representable by a scheme if there exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and an equivalence $j:\mathcal{X}\rightarrow (\text{Sch}/U)_{\text{fppf}}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$.

A category fibered in groupoids $p : \mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$ is representable by an algebraic space over $S$ if there exists an algebraic space $F$ over $S$ and an equivalence $j:\mathcal{X}\rightarrow \mathcal{S}_{F}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$.

Next, following what we did earlier for the case of algebraic spaces, we want to define the notion of representability (by algebraic spaces) for morphisms of categories fibered in groupoids (these are simply functors satisfying some compatibility conditions with the extra structure of the category). We will need, once again, the notion of a fiber product, this time of categories over some other fixed category.

Let $F:\mathcal{X}\rightarrow\mathcal{S}$ and $G:\mathcal{Y}\rightarrow\mathcal{S}$ be morphisms of categories over $\mathcal{C}$. The fiber product $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is given by the following description:

(1) an object of $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is a quadruple $(U,x,y,f)$, where $U\in\text{Ob}(\mathcal{C})$, $x\in\text{Ob}(\mathcal{X}_{U})$, $y\in\text{Ob}(\mathcal{Y}_{U})$, and $f : F(x)\rightarrow G(y)$ is an isomorphism in $\mathcal{S}_{U}$,

(2) a morphism $(U,x,y,f) \rightarrow (U',x',y',f')$ is given by a pair $(a,b)$, where $a:x\rightarrow x'$ is a morphism in $X$, and $b:y\rightarrow y'$ is a morphism in $Y$ such that $a$ and $b$ induce the same morphism $U\rightarrow U'$, and $f'\circ F(a)=G(b)\circ f$.

Let $S$ be a scheme. A morphism $f:\mathcal{X}\rightarrow \mathcal{Y}$ of categories fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ is called representable by algebraic spaces if for any $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and any $y:(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{Y}$ the category fibered in groupoids

$\displaystyle (\text{Sch}/U)_{\text{fppf}}\times_{y,\mathcal{Y}}\mathcal{X}$

over $(\text{Sch}/U)_{\text{fppf}}$ is representable by an algebraic space over $U$.

An algebraic stack (or Artin stack) over a scheme $S$ is a category

$\displaystyle p:\mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$

with the following properties:

(1) The category $\mathcal{X}$ is a stack in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ .

(2) The diagonal $\Delta:\mathcal{X}\rightarrow \mathcal{X}\times\mathcal{X}$ is representable by algebraic spaces.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch/S})_{\text{fppf}})$ and a morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ which is surjective and smooth (This is often written simply as $U\rightarrow\mathcal{X}$). Again, the scheme $U$ is called an atlas.

If the morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is surjective and etale, we have a Deligne-Mumford stack.

Just as an algebraic space is a generalization of the notion of a scheme, an algebraic stack is also a generalization of the notion of an algebraic space (recall that that a presheaf can be thought of as category fibered in sets, which themselves are special cases of groupoids). Therefore, the definition of an algebraic stack closely resembles the definition of an algebraic space given earlier, including the requirement that the diagonal morphism (which in this case is a functor between categories) be representable, so that the functor $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is also representable, and we can describe it as being surjective and smooth (or surjective and etale).

As an example of an application of the ideas just discussed, we mention the moduli stack of elliptic curves (which we denote by $\mathcal{M}_{1,1}$ – the reason for this notation will become clear later). A family of elliptic curves over some “base space” $B$ is a fibration $\pi:X\rightarrow B$ with a section $O:B\rightarrow X$ such that the fiber $\pi^{-1}(b)$ over any point $b$ of $B$ is an elliptic curve with origin $O(b)$.

Ideally what we want is to be able to obtain every family $X\rightarrow B$ by pulling back a “universal object” $E\rightarrow\mathcal{M}_{1,1}$ via the map $B\rightarrow\mathcal{M}_{1,1}$. This is something that even the notion of moduli space that we discussed in The Moduli Space of Elliptic Curves cannot do (we suggestively denote that moduli space by $M_{1,1}$). So we need the concept of stacks to construct this “moduli stack” that has this property. A more thorough discussion would need the notion of quotient stacks and orbifolds, but we only mention that the moduli stack of elliptic curves is in fact a Deligne-Mumford stack.

More generally, we can construct the moduli stack of curves of genus $g$ with $\nu$ marked points, denoted $\mathcal{M}_{g,\nu}$. The moduli stack of elliptic curves is simply the special case $\mathcal{M}_{1,1}$. Aside from just curves of course, we can construct moduli stacks for many more mathematical objects, such subschemes of some fixed scheme, or vector bundles, also on some fixed scheme.

The subject of algebraic stacks is a vast one, as may perhaps be inferred from the size of one of the main references for this post, the open-source reference The Stacks Project, which consists of almost 6,000 pages at the time of this writing. All that has been attempted in this post is but an extremely “bare bones” introduction to some of its more basic concepts. Hopefully more on stacks will be featured in future posts on the blog.

References:

Stack on Wikipedia

Algebraic Space on Wikipedia

Fibred Category on Wikipedia

Descent Theory on Wikipedia

Stack on nLab

Grothendieck Fibration on nLab

Algebraic Space on nLab

Algebraic Stack on nLab

Moduli Stack of Elliptic Curves on nLab

Stacks for Everybody by Barbara Fantechi

What is…a Stack? by Dan Edidin

Notes on the Construction of the Moduli Space of Curves by Dan Edidin

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory by Angelo Vistoli

Lectures on Moduli Spaces of Elliptic Curves by Richard Hain

The Stacks Project

Algebraic Spaces and Stacks by Martin Olsson

Fundamental Algebraic Geometry: Grothendieck’s FGA Explained by Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli

# The Theory of Motives

The theory of motives originated from the observation, sometime in the 1960’s, that in algebraic geometry there were several different cohomology theories (see Homology and Cohomology and Cohomology in Algebraic Geometry), such as Betti cohomology, de Rham cohomology, $l$-adic cohomology, and crystalline cohomology. The search for a “universal cohomology theory”, such that all these other cohomology theories could be obtained from such a universal cohomology theory is what led to the theory of motives.

The four cohomology theories enumerated above are examples of what is called a Weil cohomology theory. A Weil cohomology theory, denoted $H^{*}$, is a functor (see Category Theory) from the category $\mathcal{V}(k)$ of smooth projective varieties over some field $k$ to the category $\textbf{GrAlg}(K)$ of graded $K$-algebras, for some other field $K$ which must be of characteristic zero, satisfying the following axioms:

(1) (Finite-dimensionality) The homogeneous components $H^{i}(X)$ of $H^{*}(X)$ are finite dimensional for all $i$, and $H^{i}(X)=0$ whenever $i<0$ or $i>2n$, where $n$ is the dimension of the smooth projective variety $X$.

(2) (Poincare duality) There is an orientation isomorphism $H^{2n}\cong K$, and a nondegenerate bilinear pairing $H^{i}(X)\times H^{2n-i}(X)\rightarrow H^{2n}\cong K$.

(3) (Kunneth formula) There is an isomorphism

$\displaystyle H^{*}(X\times Y)\cong H^{*}(X)\otimes H^{*}(Y)$.

(4) (Cycle map) There is a mapping $\gamma_{X}^{i}$ from $C^{i}(X)$, the abelian group of algebraic cycles of codimension $i$ on $X$ (see Algebraic Cycles and Intersection Theory), to $H^{i}(X)$, which is functorial with respect to pullbacks and pushforwards, has the multiplicative property $\gamma_{X\times Y}^{i+j}(Z\times W)=\gamma_{X}^{i}(Z)\otimes \gamma_{Y}^{j}(W)$, and such that $\gamma_{\text{pt}}^{i}$ is the inclusion $\mathbb{Z}\hookrightarrow K$.

(5) (Weak Lefschetz axiom) If $W$ is a smooth hyperplane section of $X$, and $j:W\rightarrow X$ is the inclusion, the induced map $j^{*}:H^{i}(X)\rightarrow H^{i}(W)$ is an isomorphism for $i\leq n-2$, and a monomorphism for $i\leq n-1$.

(6) (Hard Lefschetz axiom) The Lefschetz operator

$\displaystyle \mathcal{L}:H^{i}(X)\rightarrow H^{i+2}(X)$

given by

$\displaystyle \mathcal{L}(x)=x\cdot\gamma_{X}^{1}(W)$

for some smooth hyperplane section $W$ of $X$, with the product $\cdot$ provided by the graded $K$-algebra structure of $H^{*}(X)$, induces an isomorphism

$\displaystyle \mathcal{L}^{i}:H^{n-i}(X)\rightarrow H^{n+i}(X)$.

The idea behind the theory of motives is that all Weil cohomology theories should factor through a “category of motives”, i.e. any Weil cohomology theory

$\displaystyle H^{*}: \mathcal{V}(k)\rightarrow \textbf{GrAlg}(K)$

can be expressed as the following composition of functors:

$\displaystyle H^{*}: \mathcal{V}(k)\xrightarrow{h} \mathcal{M}(k)\rightarrow\textbf{GrAlg}(K)$

where $\mathcal{M}(k)$ is the category of motives. We can get different Weil cohomology theories, such as Betti cohomology, de Rham cohomology, $l$-adic cohomology, and crystalline cohomology, via different functors (called realization functors) from the category of motives to a category of graded algebras over some field $K$. This explains the term “motive”, which actually comes from the French word “motif”, which itself is already used in music and visual arts, among other things, as some kind of common underlying “theme” with different possible manifestations.

Let us now try to construct this category of motives. This category is often referred to in the literature as a “linearization” of the category of smooth projective varieties. This means that we obtain it from some sense starting with the category of smooth projective varieties, but we also want to modify it so that it we can do linear algebra, or more properly homological algebra, in some sense. In other words, we want it to behave like the category of modules over some ring. With this in mind, we want the category to be an abelian category, so that we can make sense of notions such as kernels, cokernels, and exact sequences.

An abelian category is a category that satisfies the following properties:

(1) The morphisms form an abelian group.

(2) There is a zero object.

(3) There are finite products and coproducts.

(4) Every morphism $f:X\rightarrow Y$ has a kernel and cokernel, and satisfies a decomposition

$\displaystyle K\xrightarrow{k} X\xrightarrow{i} I\xrightarrow{j} Y\xrightarrow{c} K'$

where $K$ is the kernel of $f$, $K'$ is the cokernel of $f$, and $I$ is the kernel of $c$ and the cokernel of $k$ (not to be confused with our notation for fields).

In order to proceed with our construction of the category of motives, which we now know we want to be an abelian category, we discuss the notion of correspondences.

The group of correspondences of degree $r$ from a smooth projective variety $X$ to another smooth projective variety $Y$, written $\text{Corr}^{r}(X,Y)$, is defined to be the group of algebraic cycles of $X\times Y$ of codimension $n+r$, where $n$ is the dimension of $X$, i.e.

$\text{Corr}^{r}(X,Y)=C^{n+r}(X\times Y)$

A morphism (of varieties, in the usual sense) $f:Y\rightarrow X$ determines a correspondence from $X$ to $Y$ of degree $0$ given by the transpose of the graph of $f$ in $X\times Y$. Therefore we may think of correspondences as generalizations of the usual concept of morphisms of varieties.

As we have learned in Algebraic Cycles and Intersection Theory, whenever we are dealing with algebraic cycles, it is often useful to consider them only up to some equivalence relation. In the aforementioned post we introduced the notion of rational equivalence. This time we consider also homological equivalence and numerical equivalence between algebraic cycles.

We say that two algebraic cycles $Z_{1}$ and $Z_{2}$ are homologically equivalent if they have the same image under the cycle map, and we say that they are numerically equivalent if the intersection numbers $Z_{1}\cdot Z$ and $Z_{2}\cdot Z$ are equal for all $Z$ of complementary dimension. There are other such equivalence relations on algebraic cycles, but in this post we will only mostly be using rational equivalence, homological equivalence, and numerical equivalence.

Since correspondences are algebraic cycles, we often consider them only up to these equivalence relations, and denote the quotient group we obtain by $\text{Corr}_{\sim}^{r}(X,Y)$, where $\sim$ is the equivalence relation imposed, for example, for numerical equivalence we write $\text{Corr}_{\text{num}}^{r}(X,Y)$.

Taking the tensor product of the abelian group $\text{Corr}_{\sim}^{r}(X,Y)$ with the rational numbers $\mathbb{Q}$, we obtain the vector space

$\displaystyle \text{Corr}_{\sim}^{r}(X,Y)_{\mathbb{Q}}=\text{Corr}_{\sim}^{r}(X,Y)\otimes_{\mathbb{Z}}\mathbb{Q}$

To obtain something closer to an abelian category (more precisely, we will obtain what is known as a pseudo-abelian category, but in the case where the equivalence relation is numerical equivalence, we will actually obtain an abelian category), we need to consider “projectors”, correspondences $p$ of degree $0$ from a variety $X$ to itself such that $p^{2}=p$. So now we form a category, whose objects are $h(X,p)$ for a variety $X$ and projector $p$, and whose morphisms are given by

$\displaystyle \text{Hom}(h(X,p),h(Y,q))=q\circ\text{Corr}_{\sim}^{0}(X,Y)_{\mathbb{Q}}\circ p$.

We call this category the category of pure effective motives, and denote it by $\mathcal{M}_{\sim}^{\text{eff}}(k)$. The process described above is also known as passing to the pseudo-abelian (or Karoubian) envelope.

We write $h^{i}(X,p)$ for the objects of $\mathcal{M}_{\sim}^{\text{eff}}(k)$ that map to $H^{i}(X)$. In the case that $X$ is the projective line $\mathbb{P}^{1}$, and $p$ is the diagonal $\Delta_{\mathbb{P}^{1}}$, we find that

$h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=h^{0}\mathbb{P}^{1}\oplus h^{2}\mathbb{P}^{1}$

which can be rewritten also as

$\displaystyle h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=\mathbb{I}\oplus\mathbb{L}$

where $\mathbb{I}$ is the image of a point in the category of pure effective motives, and $\mathbb{L}$ is known as the Lefschetz motive. It is also denoted by $\mathbb{Q}(-1)$. The above decomposition corresponds to the projective line $\mathbb{P}^{1}$ being a union of the affine line $\mathbb{A}^{1}$ and a “point at infinity”, which we may denote by $\mathbb{A}^{0}$:

$\displaystyle \mathbb{P}^{1}=\mathbb{A}^{0}\cup\mathbb{A}^{1}$

More generally, we have

$\displaystyle h(\mathbb{P}^{n},\Delta_{\mathbb{P}^{n}})=\mathbb{I}\oplus\mathbb{L}\oplus...\oplus\mathbb{L}^{n}$

corresponding to

$\displaystyle \mathbb{P}^{n}=\mathbb{A}^{0}\cup\mathbb{A}^{1}\cup...\cup\mathbb{A}^{n}$.

The category of effective pure motives is an example of a tensor category. This means it has a bifunctor $\otimes: \mathcal{M}_{\sim}^{\text{eff}}\times\mathcal{M}_{\sim}^{\text{eff}}\rightarrow\mathcal{M}_{\sim}^{\text{eff}}$ which generalizes the usual notion of a tensor product, and in this particular case it is given by taking the product of two varieties. We can ask for more, however, and construct a category of motives which is not just a tensor category but a rigid tensor category, which provides us with a notion of duals.

By formally inverting the Lefschetz motive (the formal inverse of the Lefschetz motive is then known as the Tate motive, and is denoted by $\mathbb{Q}(1)$), we can obtain this rigid tensor category, whose objects are triples $h(X,p,m)$, where $X$ is a variety, $e$ is a projector, and $m$ is an integer. The morphisms of this category are given by

$\displaystyle \text{Hom}(h(X,p,n),h(Y,q,m))=q\circ\text{Corr}_{\sim}^{n-m}(X,Y)_{\mathbb{Q}}\circ p$.

This category is called the category of pure motives, and is denoted by $\mathcal{M}_{\sim}(k)$. The category $\mathcal{M}_{\text{rat}}(k)$ is called the category of Chow motives, while the category $\mathcal{M}_{\text{num}}(k)$ is called the category of Grothendieck (or numerical) motives.

The category of Chow motives has the advantage that it is known to be “universal”, in the sense that every Weil cohomology theory factors through it, as discussed earlier; however, in general it is not even abelian, which is a desirable property we would like our category of motives to have. Meanwhile, the category of Grothendieck motives is known to be abelian, but it is not yet known if it is universal. If the so-called “standard conjectures on algebraic cycles“, which we will enumerate below, are proved, then the category of Grothendieck motives will be known to be universal.

We have seen that the category of pure motives forms a rigid tensor category. Closely related to this concept, and of interest to us, is the notion of a Tannakian category. More precisely, a Tannakian category is a $k$-linear rigid tensor category with an exact faithful functor (called a fiber functor) to the category of finite-dimensional vector spaces over some field extension $K$ of $k$.

One of the things that makes Tannakian categories interesting is that there is an equivalence of categories between a Tannakian category $\mathcal{C}$ and the category $\text{Rep}_{G}$ of finite-dimensional linear representations of the group of automorphisms of its fiber functor, which is also known as the Tannakian Galois group, or, if the Tannakian category is a “category of motives” of some sort, the motivic Galois group. This aspect of Tannakian categories may be thought of as a higher-dimensional analogue of the classical theory of Galois groups, which can be stated as an equivalence of categories between the category of finite separable field extensions of a field $k$ and the category of finite sets equipped with an action of the Galois group $\text{Gal}(\bar{k}/k)$, where $\bar{k}$ is the algebraic closure of $k$.

So we see that being a Tannakian category is yet another desirable property that we would like our category of motives to have. For this not only do we have to tweak the tensor product structure of our category, we also need certain conjectural properties to hold. These are the same conjectures we have hinted at earlier, called the standard conjectures on algebraic cycles, formulated by Alexander Grothendieck at around the same time he initially developed the theory of motives.

These conjectures have some very important consequences in algebraic geometry, and while they remain unproved to this day, the search for their proof (or disproof) is an important part of modern mathematical research on the theory of motives. They are the following:

(A) (Standard conjecture of Lefschetz type) For $i\leq n$, the operator $\Lambda$ defined by

$\displaystyle \Lambda=(\mathcal{L}^{n-i+2})^{-1}\circ\mathcal{L}\circ (\mathcal{L}^{n-i}):H^{i}\rightarrow H^{i-2}$

$\displaystyle \Lambda=(\mathcal{L}^{n-i})\circ\mathcal{L}\circ (\mathcal{L}^{n-i+2})^{-1}:H^{2n-i+2}\rightarrow H^{2n-i}$

is induced by algebraic cycles.

(B) (Standard conjecture of Hodge type) For all $i\leq n/2$, the pairing

$\displaystyle x,y\mapsto (-1)^{i}(\mathcal{L}x\cdot y)$

is positive definite.

(C) (Standard conjecture of Kunneth type) The projectors $H^{*}(X)\rightarrow H^{i}(X)$ are induced by algebraic cycles in $X\times X$ with rational coefficients. This implies the following decomposition of the diagonal:

$\displaystyle \Delta_{X}=\pi_{0}+...+\pi_{2n}$

which in turn implies the decomposition

$\displaystyle h(X,\Delta_{X},0)=h(X,\pi_{0},0)\oplus...\oplus h(X,\pi_{2n},0)$

which, writing $h(X,\Delta_{X},0)$ as $hX$ and $h(X,\pi_{i},0)$ as $h^{i}(X)$, we can also compactly and suggestively write as

$\displaystyle hX=h^{0}X\oplus...\oplus h^{2n}X$.

In other words, every object $hX=h(X,\Delta_{X},0)$ of our “category of motives” decomposes into graded “pieces” $h^{i}(X)=h(X,\pi_{i},0)$ of pure “weight$i$. We have already seen earlier that this is indeed the case when $X=\mathbb{P}^{n}$. We will need this conjecture to hold if we want our category to be a Tannakian category.

(D) (Standard conjecture on numerical equivalence and homological equivalence) If an algebraic cycle is numerically equivalent to zero, then its cohomology class is zero. If the category of Grothendieck motives is to be “universal”, so that every Weil cohomology theory factors through it, this conjecture must be satisfied.

In Algebraic Cycles and Intersection Theory and Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry, we have made mention of the two famous conjectures in algebraic geometry known as the Hodge conjecture and the Tate conjecture. In fact, these two closely related conjectures can be phrased in the language of motives as the conjectures stating that the realization functors from the category of motives to the category of pure Hodge structures and continuous $l$-adic representations of $\text{Gal}(\bar{k}/k)$, respectively, be fully faithful. These conjectures are closely related to the standard conjectures on algebraic cycles as well.

We have now constructed the category of pure motives, for smooth projective varieties. For more general varieties and schemes, there is an analogous idea of “mixed motives“, which at the moment remain conjectural, although there exist several related constructions which are the closest thing we currently have to such a theory of mixed motives.

If we want to construct a theory of mixed motives, instead of Weil cohomology theories we must instead consider what are known as “mixed Weil cohomology theories“, which are expected to have the following properties:

(1) (Homotopy invariance) The projection $\pi:X\rightarrow\mathbb{A}^{1}$ induces an isomorphism

$\displaystyle \pi^{*}:H^{*}(X)\xrightarrow{\cong}H^{*}(X\times\mathbb{A}^{1})$

(2) (Mayer-Vietoris sequence) If $U$ and $V$ are open coverings of $X$, then there is a long exact sequence

$\displaystyle ...\rightarrow H^{i}(U\cap V)\rightarrow H^{i}(X)\rightarrow H^{i}(U)\oplus H^{i}(V)\rightarrow H^{i}(U\cap V)\rightarrow...$

(3) (Duality) There is a duality between cohomology $H^{*}$ and cohomology with compact support $H_{c}^{*}$.

(4) (Kunneth formula) This is the same axiom as the one in the case of pure motives.

We would like a category of mixed motives, which serves as an analogue to the category of pure motives in that all mixed Weil cohomology theories factor through it, but as mentioned earlier, no such category exists at the moment. However, the mathematicians Annette Huber-Klawitter, Masaki Hanamura, Marc Levine, and Vladimir Voevodsky have constructed different versions of a triangulated category of mixed motives, denoted $\mathcal{DM}(k)$.

A triangulated category $\mathcal{T}$ is an additive category with an automorphism $T: \mathcal{T}\rightarrow\mathcal{T}$ called the “shift functor” (we will also denote $T(X)$ by $X[1]$, and $T^{n}(X)$ by $X[n]$, for $n\in\mathbb{Z}$) and a family of “distinguished triangles

$\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]$

which satisfies the following axioms:

(1) For any object $X$ of $\mathcal{T}$, the triangle $X\xrightarrow{\text{id}}X\rightarrow 0\rightarrow X[1]$ is a distinguished triangle.

(2) For any morphism $u:X\rightarrow Y$ of $\mathcal{T}$, there is an object $Z$ of $\mathcal{T}$ such that $X\xrightarrow{u}Y\rightarrow Z\rightarrow X[1]$ is a distinguished triangle.

(3) Any triangle isomorphic to a distinguished triangle is a distinguished triangle.

(4) If $X\rightarrow Y\rightarrow Z\rightarrow X[1]$ is a distinguished triangle, then the two “rotations” $Y\rightarrow Z\rightarrow Z[1]\rightarrow Y[1]$ and $Z[-1]\rightarrow X\rightarrow Y\rightarrow Z$ are also distinguished triangles.

(5) Given two distinguished triangles $X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]$ and $X'\xrightarrow{u'}Y'\xrightarrow{v'}Z'\xrightarrow{w'}X'[1]$ and morphisms $f:X\rightarrow X'$ an $g:Y\rightarrow Y'$ such that the square “commutes”, i.e. $u'\circ f=g\circ u$, there exists a morphisms $h:Z\rightarrow Z$ such that all other squares commute.

(6) Given three distinguished triangles $X\xrightarrow{u}Y\xrightarrow{j}Z'\xrightarrow{k}X[1]$$Y\xrightarrow{v}Z\xrightarrow{l}X'\xrightarrow{i}Y[1]$, and $X\xrightarrow{v\circ u}Z\xrightarrow{m}Y'\xrightarrow{n}X[1]$, there exists a distinguished triangle $Z'\xrightarrow{f}Y'\xrightarrow{g}X'\xrightarrow{h}Z'[1]$ such that “everything commutes”.

A $t$-structure on a triangulated category $\mathcal{T}$ is made up of two full subcategories $\mathcal{T}^{\geq 0}$ and $\mathcal{T}^{\leq 0}$ satisfying the following properties (writing $\mathcal{T}^{\leq n}$ and $\mathcal{T}^{\leq n}$ to denote $\mathcal{T}^{\leq 0}[-n]$ and $\mathcal{T}^{\geq 0}[-n]$ respectively):

(1) $\mathcal{T}^{\leq -1}\subset \mathcal{T}^{\leq 0}$ and $\mathcal{T}^{\geq 1}\subset \mathcal{T}^{\geq 0}$

(2) $\displaystyle \text{Hom}(X,Y)=0$ for any object $X$ of $\mathcal{T}^{\leq 0}$ and any object $Y$ of $\mathcal{T}^{\geq 1}$

(3) for any object $Y$ of $\mathcal{T}$ we have a distinguished triangle

$\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]$

where $X$ is an object of $\mathcal{T}^{\leq 0}$ and $Z$ is an object of $\mathcal{T}^{\geq 1}$.

The full subcategory $\mathcal{T}^{0}=\mathcal{T}^{\leq 0}\cap\mathcal{T}^{\geq 0}$ is called the heart of the $t$-structure, and it is an abelian category.

It is conjectured that the category of mixed motives $\mathcal{MM}(k)$ is the heart of the $t$-structure of the triangulated category of mixed motives $\mathcal{DM}(k)$.

Voevodsky’s construction proceeds in a manner somewhat analogous to the construction of the category of pure motives as above, starting with schemes (say, over a field $k$, although a more general scheme may be used) as objects and correspondences as morphisms, but then makes use of concepts from abstract homotopy theory, such as taking the bounded homotopy category of bounded complexes, and localization with respect to a certain subcategory, before passing to the pseudo-abelian envelope and then formally inverting the Tate object $\mathbb{Z}(1)$. The triangulated category obtained is called the category of geometric motives, and is denoted by $\mathcal{DM}_{\text{gm}}(k)$. The schemes and correspondences involved in the construction of $\mathcal{DM}_{\text{gm}}(k)$ are required to satisfy certain properties which eliminates the need to consider the equivalence relations which form a large part of the study of the category of pure motives.

Closely related to the triangulated category of mixed motives is motivic cohomology, which is defined in terms of the former as

$\displaystyle H^{i}(X,\mathbb{Z}(m))=\text{Hom}_{\mathcal{DM}(k)}(X,\mathbb{Z}(m)[i])$

where $\mathbb{Z}(m)$ is the tensor product of $m$ copies of the Tate object $\mathbb{Z}(1)$, and the notation $\mathbb{Z}(m)[i]$ tells us that the shift functor of the triangulated category is applied to the object $\mathbb{Z}(m)$ $i$ times.

Motivic cohomology is related to the Chow group, which we have introduced in Algebraic Cycles and Intersection Theory, and also to algebraic K-theory, which is another way by which the ideas of homotopy theory are applied to more general areas of abstract algebra and linear algebra. These ideas were used by Voevodsky to prove several related theorems, from the Milnor conjecture to its generalization, the Bloch-Kato conjecture (also known as the norm residue isomorphism theorem).

Historically, one of the motivations for Grothendieck’s attempt to obtain a universal cohomology theory was to prove the Weil conjectures, which is a higher-dimensional analogue of the Riemann hypothesis for curves over finite fields first proved by Andre Weil himself (see The Riemann Hypothesis for Curves over Finite Fields). In fact, if the standard conjectures on algebraic cycles are proved, then a proof of the Weil conjectures would follow via an approach that closely mirrors Weil’s original proof (since cohomology provides a Lefschetz fixed-point formula –  we have mentioned in The Riemann Hypothesis for Curves over Finite Fields that the study of fixed points is an important part of Weil’s proof). The last of the Weil conjectures were eventually proved by Grothendieck’s student Pierre Deligne, but via a different approach that bypassed the standard conjectures. A proof of the standard conjectures, which would lead to a perhaps more elegant proof of the Weil conjectures, is still being pursued to this day.

The theory of motives is not only related to analogues of the Riemann hypothesis, which concerns the location of zeroes of L-functions, but to L-functions in general. For instance, it is also related to the Langlands program, which concerns another aspect of L-functions, namely their analytic continuation and functional equation, and to the Birch and Swinnerton-Dyer conjecture, which concerns their values at special points.

We recall in The Riemann Hypothesis for Curves over Finite Fields that the Frobenius morphism played an important part in counting the points of a curve over a finite field, which in turn we needed to define the zeta function (of which the L-function can be thought of as a generalization) of the curve. The Frobenius morphism is an element of the Galois group, and we recall that a category of motives which is a Tannakian category is equivalent to the category of representations of its motivic Galois group. Therefore we can see how we can define “motivic L-functions” using the theory of motives.

As the L-functions occupy a central place in many areas of modern mathematics, the theory of motives promises much to be gained from its study, if only we could make progress in deciphering the many mysteries that surround it, of which we have only scratched the surface in this post. The applications of motives are not limited to L-functions either – the study of periods, which relate Betti cohomology and de Rham cohomology, and lead to transcendental numbers which can be defined using only algebraic concepts, is also strongly connected to the theory of motives. Recent work by the mathematicians Alain Connes and Matilde Marcolli has also suggested applications to physics, particularly in relation to Feynman diagrams in quantum field theory. There is also another generalization of the theory of motives, developed by Maxim Kontsevich, in the context of noncommutative geometry.

References:

Weil Cohomology Theory on Wikipedia

Motive on Wikipedia

Standard Conjectures on Algebraic Cycles on Wikipedia

Motive on nLab

Pure Motive on nLab

Mixed Motive on nLab

The Tate Conjecture over Finite Fields on Hard Arithmetic

What is…a Motive? by Barry Mazur

Motives – Grothendieck’s Dream by James S. Milne

Noncommutative Geometry, Quantum Fields, and Motives by Alain Connes and Matilde Marcolli

Algebraic Cycles and the Weil Conjectures by Steven L. Kleiman

The Standard Conjectures by Steven L. Kleiman

Feynman Motives by Matilde Marcolli

Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Periodes) by Yves Andre

# Some Useful Links on the History of Algebraic Geometry

It’s been a while since I’ve posted on this blog, but there are some posts I’m currently working on about some subjects I’m also currently studying (that’s why it’s taking so long, as I’m trying to digest the ideas as much as I can before I can post about it). But anyway, for the moment, in this short post I’ll be putting up some links to articles on the history of algebraic geometry. Aside from telling an interesting story on its own, there is also much to be learned about a subject from studying its historical development.

We know that the origins of algebraic geometry can be traced back to Rene Descartes and Pierre de Fermat in the 17th century. This is the high school subject also known as “analytic geometry” (which, as we have mentioned in Basics of Algebraic Geometry, can be some rather confusing terminology, because in modern times the word “analytic” is usually used to refer to concepts in complex calculus).

The so-called “analytic geometry” seems to be a rather straightforward subject compared to modern-day algebraic geometry, which, as may be seen on many of the previous posts on this blog, is very abstract (but it is also this abstraction that gives it its power). How did this transformation come to be?

The mathematician Jean Dieudonne, while perhaps more known for his work in the branch of mathematics we call analysis (the more high-powered version of calculus), also served as adviser to Alexander Grothendieck, one of the most important names in the development of modern algebraic geometry. Together they wrote the influential work known as Elements de Geometrie Algebrique, often simply referred to as EGA. Dieudonne was also among the founding members of the “Bourbaki group”, a group of mathematicians who greatly influenced the development of modern mathematics. Himself a part of its development, Dieudonne wrote many works on the history of mathematics, among them the following article on the history of algebraic geometry which can be read for free on the website of the Mathematical Association of America:

The Historical Development of Algebraic Geometry by Jean Dieudonne

But before the sweeping developments instituted by Alexander Grothendieck, the modern revolution in algebraic geometry was first started by the mathematicians Oscar Zariski and Andre Weil (we discussed some of Weil’s work in The Riemann Hypothesis for Curves over Finite Fields). Zariski himself learned from the so-called “Italian school of algebraic geometry”, particularly the mathematicians Guido Castelnuovo, Federigo Enriques, and Francesco Severi.

At the International Congress of Mathematicians in 1950, both Zariski and Weil presented, separately, a survey of the developments in algebraic geometry at the time, explaining how the new “abstract algebraic geometry” was different from the old “classical algebraic geometry”, and the new advantages it presented. The proceedings of this conference are available for free online:

Proceedings of the 1950 International Congress of Mathematicians, Volume I

Proceedings of the 1950 International Congress of Mathematicians, Volume II

The articles by Weil and Zariski can be found in the second volume, but I included also the first volume for “completeness”.

All proceedings of the International Congress of Mathematicians, which is held every four years, are actually available for free online:

Proceedings of the International Congress of Mathematicians, 1983-2010

The proceedings of the 2014 International Congress of Mathematicians in Seoul, Korea, can be found here:

Proceedings of the 2014 International Congress of Mathematicians

Going back to algebraic geometry, a relatively easy to understand (for those with some basic mathematical background, anyway) summary of the work of Alexander Grothendieck’s work in algebraic geometry can be found in the following article by Colin McLarty, published in April 2016 issue of the Notices of the American Mathematical Society:

How Grothendieck Simplified Algebraic Geometry by Colin McLarty

# Tangent Spaces in Algebraic Geometry

We have discussed the notion of a tangent space in Differentiable Manifolds Revisited in the context of differential geometry. In this post we take on the same topic, but this time in the context of algebraic geometry, where it is also known as the Zariski tangent space (when no confusion arises, however, it is often simply referred to as the tangent space).

This will present us with challenges, since the concept of the tangent space is perhaps best tackled using the methods of calculus, but in algebraic geometry, we want to have a notion of tangent spaces in cases where we would not usually think of calculus as being applicable, for instance in the case of varieties over finite fields. In other words, we want our treatment to be algebraic. Nevertheless, we will use the methods of calculus as an inspiration.

We don’t want to be too dependent on the parts of calculus that make use of properties of the real and complex numbers that will not carry over to the more general cases. Fortunately, if we are dealing with polynomials, we can just “borrow” the “power rule” of calculus, since that “rule” only makes use of algebraic procedures, and we need not make use of sequences, limits, and so on. Namely, if we have a polynomial given by

$\displaystyle f=\sum_{j=1}^{n}ax^{j}$

We set

$\displaystyle \frac{\partial f}{\partial x}=\sum_{j=1}^{n}jax^{j-1}$

We recall the rules for partial derivatives – in the case that we are differentiating over some variable $x$, we simply treat all the other variables as constants, and follow the usual rules of differential calculus. With these rules, we can now make the definition of the tangent space at the point $P$ with coordinates $(a_{1},a_{2},...,a_{n})$ as the algebraic set which satisfies the equation

$\displaystyle \sum_{j}\frac{\partial f}{\partial x_{j}}(P)(x_{j}-a_{j})=0$

For example, consider the parabola given by the equation $y-x^{2}=0$. Let us take the tangent space at the point $P$ with coordinates $x=1$, $y=1$. The procedure above gives us

$\displaystyle \frac{\partial f}{\partial x}(P)(x-1)+\frac{\partial f}{\partial y}(P)(y-1)=0$

Since

$\displaystyle \frac{\partial f}{\partial x}=-2x$

$\displaystyle \frac{\partial f}{\partial y}=1$

We then have

$\displaystyle -2x|_{x=1,y=1}(x-1)+1|_{x=1,y=1}(y-1)=0$

$\displaystyle -2(1)(x-1)+1(y-1)=0$

$\displaystyle -2x+2+y-1=0$

$\displaystyle y-2x+1=0$

The parabola is graphed (its real part, at least, using the Desmos graphing calculator) in the diagram below in red, with its tangent space, a line, in blue:

In case the reader is not convinced by our “borrowing” of concepts from calculus and claiming that they are “algebraic” in the specific case we are dealing with, another way to look at things without making reference to calculus is the following procedure, which comes from basic high school-level “analytic geometry”. First we translate the coordinate system so that the origin is at the point $P$ where we want to take the tangent space. Then we simply take the “linear part” of the polynomial equation, then translate again so that the origin is where it used to be originally. This gives the same results as the earlier procedure (the technical justification is given by the theory of Taylor series). More explicitly we have:

$\displaystyle y-x^{2}=0$

Translating the origin of coordinates to the point $x=1$, $y=1$, we have

$\displaystyle (y+1)-(x+1)^{2}=0$

$\displaystyle y+1-(x^{2}+2x+1)=0$

$\displaystyle y+1-x^{2}-2x-1=0$

$\displaystyle y-x^{2}-2x=0$

We take only the linear part, which is

$\displaystyle y-2x=0$

And then we translate the origin of coordinates back to the original one:

$\displaystyle (y-1)-2(x-1)=0$

$\displaystyle y-1-2x+2=0$

$\displaystyle y-2x+1=0$

which is the same result we had earlier.

But it may happen that the polynomial has no “linear part”. In this case the tangent space is the entirety of the ambient space. However, there is another related concept which may be useful in these cases, called the tangent cone. The tangent cone is the algebraic set which satisfies the equations we get by extracting the lowest degree part of the polynomial, which may or may not be the linear part. In the case that the lowest degree part is the linear part, the tangent space and the tangent cone coincide, and if this holds for all points of a variety, we say that the variety is nonsingular.

To give an explicit example, consider the curve $y^{2}=x^{3}+x^{2}$, as seen in the diagram below in red (its real part graphed once again using the Desmos graphing calculator):

The equation that defines this curve has no linear part. Therefore the tangent space at the origin consists of all $x$ and $y$ which satisfy the trivial equation $0=0$; but then, all values of $x$ and $y$ satisfy this equation, and therefore the tangent space is the “affine plane” $\mathbb{A}^{2}$. However, the lowest order part is $y^{2}=x^{2}$, which is satisfied by all points which also satisfy either of the two equations $y=x$ or $y=-x$. These points form the blue and orange diagonal lines in the diagram. Since the tangent space and the tangent cone do not agree, the curve is singular at the origin.

We can also define the tangent space in a more abstract manner, using the concepts we have discussed in Localization. Let $\mathfrak{m}$ be the unique maximal ideal of the local ring $O_{X,P}$, and let $\mathfrak{m}^{2}$ be the product ideal whose elements are the sums of products of elements of $\mathfrak{m}$. The quotient $\mathfrak{m}/\mathfrak{m}^{2}$ is then a vector space over the residue field $k$. The tangent space of $X$ at $P$ is then defined as the dual of this vector space (the vector space of linear transformations from $\mathfrak{m}/\mathfrak{m}^{2}$ to $k$). The vector space $\mathfrak{m}/\mathfrak{m}^{2}$ itself is called the cotangent space of $X$ at $P$. We can think of its elements as linear polynomial functions on the tangent space. There is an analogous abstract definition of the tangent cone, namely as the spectrum of the graded ring $\oplus_{i\geq 0}\mathfrak{m}^{i}/\mathfrak{m}^{i+1}$.

References:

Zariski Tangent Space on Wikipedia

Tangent Cone on Wikipedia

Desmos Graphing Calculator

Algebraic Geometry by J.S. Milne

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne

# Grothendieck’s Relative Point of View

In Varieties and Schemes Revisited we defined the notion of schemes, which is a far-reaching generalization inspired by the concept of varieties, which is essentially a kind of “shape” defined by polynomials in some way. However, the definition of schemes were but one of many innovations in algebraic geometry developed by the mathematician Alexander Grothendieck. In this post, we discuss another of these innovations, the so-called “relative point of view“, in which the focus is not just on schemes in isolation, but schemes relative to (with a morphism to) some “base scheme”.

Let $S$ be a scheme. A scheme over $S$, or an $S$-scheme, is a scheme $X$ with a morphism $f:X\rightarrow S$ called the structural morphism. If $Y$ is another $S$-scheme with structural morphism $g:Y\rightarrow S$, a morphism of $S$-schemes is a morphism $u:X\rightarrow Y$ such that $f=g\circ u$.

If the scheme $S$ is the spectrum of some ring $R$, we may also refer to $X$ above as a scheme over $R$. Every ring has a morphism from the ring of ordinary integers $\mathbb{Z}$, and every scheme therefore has a morphism to the scheme $\text{Spec}(\mathbb{Z})$, so we may think of all schemes as schemes over $\mathbb{Z}$.

Given two schemes $X$ and $Y$ over a third scheme $S$, we define the fiber product $X\times_{S}Y$ to be a scheme together with projection morphisms $\pi_{X}:X\times_{S}Y\rightarrow X$ and $\pi_{Y}:X\times_{S}Y\rightarrow Y$ such that $f\circ\pi_{X}=g\circ\pi_{Y}$, and such that for any other scheme $Z$ and morphisms $p:Z\rightarrow X$ and $q:Z\rightarrow Y$, there is a unique morphism $Z\rightarrow X\times_{S}Y$ up to isomorphism (the concept of fiber product is part of category theory – see also More Category Theory: The Grothendieck Topos).

We can use the fiber product to introduce the concept of base change. Given a scheme $X$ over a scheme $S$, and a morphism $S'\rightarrow S$, the fiber product $X\times_{S}S'$ is a scheme over $S'$. We may think of it as being “induced” by the morphism $S'\rightarrow S$. One of the things that can be done with this idea of base change is to look at the properties of $X\times_{S}S'$ and see if we can use these to learn about the properties of $X$, which may be useful if the properties of $X$ are difficult to determine directly compared to the properties of $X\times_{S}S'$ (in essence we want to be able to attack a difficult problem indirectly by first attacking an easier problem related to it, which is a common strategy in mathematics).

A special case of base change is when $S'$ is given by the spectrum of the residue field (see Localization) $k$ corresponding to a point $P$ of $S$. There is a morphism of schemes $\text{Spec}(k)\rightarrow S$ which we may think of as the inclusion of the point $P$ into the scheme $X$. Then the fiber product $X\times_{S}\text{Spec}(k)$ is called the fiber of $X$ at the point $P$. The terminology is perhaps reminiscent of fiber bundles (see Vector Fields, Vector Bundles, and Fiber Bundles), and is also rather similar to the concept of covering spaces (see Covering Spaces) in that we have some kind of space “over” every point of our “base” scheme. However, unlike those two earlier concepts, the spaces which make up our fibers may now vary as the points vary.

Actually, the concept that this special case of fiber product and base change should bring to mind is that of a moduli space (see The Moduli Space of Elliptic Curves), where every point represents a space, and the spaces vary as the points vary. Or, as we worded it in The Moduli Space of Elliptic Curves, every point of the moduli space (given by the base scheme) corresponds to a space (given by the fiber), and the moduli space tells us how these spaces vary, so that spaces which are similar to each other in some way correspond to points in the moduli space that are close together.

The lecture notes of Andreas Gathmann listed among the references below contain some nice diagrams to help visualize the idea of the fiber product and base change (these can be found in chapter 5 of the 2002 version). To see these ideas in action, one can look at the article Arithmetic on Curves by Barry Mazur (also among the references) which discusses, among other things, the approach taken by Gerd Faltings in proving the famous conjecture of Louis J. Mordell which says that there is a finite number of rational points on a curve of genus greater than $1$.

References:

Grothendieck’s Relative Point of View on Wikipedia

Arithmetic on Curves by Barry Mazur

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Algebraic Geometry by Robin Hartshorne

# Varieties and Schemes Revisited

In Basics of Algebraic Geometry we introduced the idea of varieties and schemes as being kinds of “shapes” defined by polynomials (or rings, more generally) in some way. In this post we discuss the definitions of these concepts in more technical detail, and introduce other important concepts related to algebraic geometry as well.

##### I. Preliminaries: Affine Space, Algebraic Sets and Ringed Spaces

Affine $n$-space, written $\mathbb{A}^{n}$, is the set of all $n$-tuples of elements of a field $k$, i.e.

$\displaystyle \mathbb{A}^{n}=\{(a_{1},...,a_{n})|a_{i}\in k \text{ for }1\leq i\leq n\}$.

An algebraic set is a subset of $\mathbb{A}^{n}$ that is the zero set $Z(T)$ of some set $T$ of polynomials, i.e. $Y=Z(T)$, where

$\displaystyle Z(T)=\{P\in \mathbb{A}^{n}|f(P)=0 \text{ for all } f\in T\}$.

Intuitively, we want to define a “variety” as some kind of space which “locally” looks like an irreducible algebraic set. “Irreducible” means it cannot be expressed as the union of other algebraic sets. However, we want to think of a variety as more than just a space; we want to think of it as a space with things (namely functions) “living on it”. This leads us to the notion of a ringed space.

A ringed space is simply a pair $(X,\mathcal{O}_{X})$, where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf (see  Sheaves) of rings on $X$. A morphism of ringed spaces from $(X,O_{X})$ to $(Y,O_{Y})$ is given by a continuous map $f: X\rightarrow Y$ and a morphism of sheaves of rings $f^{\#}: \mathcal{O}_{Y}\rightarrow f_{*}\mathcal{O}_{X}$.

Recall that a morphism of sheaves of rings $\varphi:\mathcal{F}\rightarrow \mathcal{G}$ for sheaves of rings $\mathcal{F}$ and $\mathcal{G}$ on X is given by a morphism of rings $\varphi(U): \mathcal{F}(U)\rightarrow \mathcal{G}(U)$ for every open set $U$ of $X$ such that for $V\subseteq{U}$ we have $\rho_{U,V}\circ\varphi(U)=\varphi(V)\circ\rho'_{U,V}$, where $\rho_{U,V}$ and $\rho'_{U,V}$ are the restriction maps of $\mathcal{F}$ and $\mathcal{G}$.

We might as well mention locally ringed spaces here, since they will be used to define the concept of schemes later on:

A locally ringed space is a ringed space $(X,\mathcal{O}_{X})$ such that for each point $P$ of $X$, the stalk $\mathcal{O}_{X,P}$ is a local ring (see Localization). A morphism of locally ringed spaces from $(X,O_{X})$ to $(Y,O_{Y})$ is given by a continuous map $f: X\rightarrow Y$ and a morphism of sheaves of rings $f^{\#}: \mathcal{O}_{Y}\rightarrow f_{*}\mathcal{O}_{X}$ such that $(f_{P}^{\#})^{-1}(\mathfrak{m}_{X,P})=\mathfrak{m}_{Y,f(P)}$ for all $P$ where $f_{P}^{\#}: \mathcal{O}_{Y,f(P)}\rightarrow \mathcal{O}_{X,P}$ is the map induced on the stalk at $P$.

##### II. Varieties in Three Steps:  Affine Varieties, Prevarieties, and Varieties

We now set out to accomplish our goal of defining “varieties” as spaces that locally look like irreducible algebraic sets. We first start with a ringed space that just “looks like” an irreducible algebraic set:

An affine variety is a ringed space $(X,\mathcal{O}_{X})$ such that $X$ is irreducible, $O_{X}$ is a sheaf of $k$-valued functions, and $X$ is isomorphic to an irreducible algebraic set in $\mathbb{A}^{n}$.

Next, we define a more general kind of ringed space, that is required to look like an irreducible algebraic set only “locally”:

A prevariety is a ringed space $(X,\mathcal{O}_{X})$ such that $X$ is irreducible, $O_{X}$ is a sheaf of $k$-valued functions, and there is a finite open cover $U_{i}$ such that $(U_{i},\mathcal{O}_{X}|_{U_{i}})$ is an affine variety for all $i$.

We are almost done. However, there is one more nice property that we would like our varieties to have. A topological space $X$ is said to have the Hausdorff property if two distinct points always have two disjoint neighborhoods. With the Zariski topology this is almost always impossible; however there is an analogous notion which is satisfied if the image of the “diagonal morphism” which sends the point $P$ in $X$ to the point $(P,P)$ in $X\times X$ is closed in $X\times X$. There is an analogous notion of “product” in algebraic geometry; therefore, we can define the concept of variety as follows:

A variety is a prevariety $X$ such that the diagonal morphism is closed in $X\times X$. In the rest of this post, we will refer to this property as the “algebro-geometric” analogue of the Hausdorff property.

##### III. Schemes

We now define the concept of schemes, which, as we shall show in the next section, generalize the concept of varieties, i.e. varieties are just a special case of schemes. Inspired by the correspondence between the maximal ideals of the “ring of polynomial functions” (with coefficients in an “algebraically closed field” like the complex numbers) of an algebraic set and the points of the algebraic set mentioned in Basics of Algebraic Geometry, we go further and consider a ringed space whose underlying topological space has points corresponding to the prime ideals of a ring (which is not necessarily a ring of polynomials – we might even consider, for example, the ring of ordinary integers $\mathbb{Z}$, or the ring of integers of an algebraic number field –  see Algebraic Numbers).

The spectrum (note that the word “spectrum” has many different meanings in mathematics, and this particular usage is different, say, from that in Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories) of a ring is a  locally ringed space $(\text{Spec}(A)),\mathcal{O}$, where $\text{Spec}(A)$ is the set of prime ideals of $A$ equipped with the Zariski topology, and $\mathcal O$ is a sheaf on $\text{Spec}(A)$ given by defining $\mathcal{O}(U)$ to be the set of functions $s:U\rightarrow \coprod_{\mathfrak{p}\in U}A_{\mathfrak{p}}$, such that $s(\mathfrak{p})\in A_\mathfrak{p}$ for each $\mathfrak{p}\in U$, and such that for each $\mathfrak{p}\in U$, there is an open set $V\subseteq U$ containing $\mathfrak{p}$ and elements $a,f\in A$ such that for each $\mathfrak{q}\in V$, $f\notin \mathfrak{q}$, and $s(\mathfrak{q})=a/f$ in $A_{\mathfrak{q}}$.

We now proceed to define schemes, closely mirroring how we defined varieties earlier:

An affine scheme is a locally ringed space $(X,\mathcal{O}_{X})$ that is isomorphic as a locally ringed space to the spectrum of some ring.

A scheme is a locally ringed space $(X,\mathcal{O}_{X})$ where every point is contained in some open set $U$ such that $U$ considered as a topological space, together with the restricted sheaf $\mathcal{O}_{X}|_{U}$, is an affine scheme. A morphism of schemes is a morphism as locally ringed spaces.

Finally, to complete the analogy with varieties, we refer to schemes which have the (analogue of the) Hausdorff property as separated schemes.

Note: In some of the (mostly older) literature, what we refer to as schemes in this post are instead referred to as preschemes, in analogy with prevarieties. What they call a scheme is what we refer to as a separated scheme, i.e. a scheme possessing the Hausdorff property. I have no idea at the moment as to why this rather nice terminology was changed, but in this post we stick with the modern convention.

##### IV. Prevarieties and Varieties as Special Kinds of Schemes

We now discuss varieties as special cases of schemes. First we need to define what properties we would like our schemes to have, in order to fit with how we described varieties earlier (as ringed spaces which locally look like irreducible spaces defined by polynomials). Therefore, we have to mimic certain properties of polynomial rings.

We first note that polynomials over a field are finitely generated algebras over some field $k$. A scheme is said to be of finite type over the field $k$ if the affine open sets are each isomorphic to the spectrum of some ring which is a finitely generated algebra over $k$. More generally, given a morphism of schemes $X\rightarrow Y$, there is a concept of $X$ being a scheme of finite type over $Y$, but we will leave this to the references for now.

Next we note that polynomials over a field are integral domains. This means that whenever there are two polynomials $f$ and $g$ with the property that $fg=0$, then either $f=0$ or $g=0$. A scheme is integral if each the affine open sets are each isomorphic to the spectrum of some ring which is an integral domain. An equivalent condition is for the scheme to be irreducible and reduced (this means that the ring specified above has no nilpotent elements, i.e. elements where some power is equal to zero).

We therefore redefine a prevariety as an integral scheme of finite type over the field $k$. As with the earlier definition, a variety is a prevariety with the (analogue of the) Hausdorff property (i.e. an integral separated scheme of finite type over $k$).

##### V. Conclusion

In conclusion, we have started with essentially the same ideas as the “analytic geometry” of Pierre de Fermat and Rene Descartes, familiar to high school students everywhere, used to describe shapes such as lines, circles, conics (parabolas, hyperbolas, circles, and ellipses), and so on. From there we generalized to get more shapes, which resemble only these old shapes “locally” (we may also think of these new shapes as being “glued” from the old ones). To maintain certain familiar properties expected of shapes, we impose the analogue of the Hausdorff property. We then obtain the concept of a variety.

But we can generalize much, much farther to more than just polynomial rings. We can define “spaces” which come from rings which need not be polynomial rings, such as the ring of ordinary integers $\mathbb{Z}$ (or more generally algebraic integers – we have actually hinted at these applications of algebraic geometry in Divisors and the Picard Group). We can then have a kind of “geometry” of these rings, which gives us methods analogous to the powerful methods of geometry, which can be applied to branches of mathematics we would not usually think of as being “geometric”, such as number theory, as we have mentioned above. We end this post with quotes from two of the pioneers of modern mathematics (these quotes are also found in the book Algebra by Michael Artin):

“To me algebraic geometry is algebra with a kick.”

-Solomon Lefschetz

“In helping geometry, modern algebra is helping itself above all.”

-Oscar Zariski

References:

Algebraic Variety on Wikipedia

Scheme on Wikipedia

Ringed Space on Wikipedia

Abstract Varieties on Rigorous Trivialities

Schemes on Rigorous Trivialities

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Algebraic Geometry by Robin Hartshorne

Algebra by Michael Artin

# Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry

I’m going to be fairly busy in the coming days, so instead of the usual long post, I’m going to post here some links to interesting stuff I’ve found online (related to the subjects stated on the title of this post).

In the previous post, An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, we discussed Calabi-Yau manifolds (which are special cases of Kahler manifolds) and how their interesting properties, namely their Riemannian, symplectic, and complex aspects figure into the branch of mathematics called mirror symmetry, which is inspired by the famous, and sometimes controversial, proposal for a theory of quantum gravity (and more ambitiously a candidate for the so-called “Theory of Everything”), string theory.

We also mentioned briefly a famous open problem concerning Kahler manifolds called the Hodge conjecture (which was also mentioned in Algebraic Cycles and Intersection Theory). The links I’m going to provide in this post will be related to this conjecture.

As with the post An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, aside from introducing the subject itself, another of the primary intentions will be to motivate and explore aspects of algebraic geometry such as complex algebraic geometry, and their relation to other branches of mathematics.

Here is the page on the Hodge conjecture, found on the website of the Clay Mathematics Institute:

Hodge Conjecture on Clay Mathematics Institute

We have mentioned before that the Hodge conjecture is one of seven “Millenium Problems” for which the Clay Mathematics Institute is offering a million dollar prize. The page linked to above contains the official problem statement as stated by Pierre Deligne, and a link to a lecture by Dan Freed, which is intended for a general audience and quite understandable. The lecture by Freed is also available on Youtube:

Dan Freed on the Hodge Conjecture at the Clay Mathematics Institute on Youtube

Unfortunately the video of that lecture has messed up audio (although the lecture remains understandable – it’s just that the audio comes out of only one side of the speakers or headphones). Here is another set of videos by David Metzler on Youtube, which explains the Hodge conjecture (along with the other Millennium Problems) to a general audience:

The Hodge conjecture is also related to certain aspects of number theory. In particular, we have the Tate conjecture, which is another conjecture similar to the Hodge conjecture, but more related to Galois groups (see Galois Groups). Alex Youcis discusses it on the following post on his blog Hard Arithmetic:

The Tate Conjecture over Finite Fields on Hard Arithmetic

On the same blog there is also a discussion of a version of the Hodge conjecture called the $p$-adic Hodge conjecture on the following post:

An Invitation to p-adic Hodge Theory; or How I Learned to Stop Worrying and Love Fontaine on Hard Arithmetic

The first part of the post linked to above discusses the Hodge conjecture in its classical form, while the second part introduces $p$-adic numbers and related concepts, some aspects of which were discussed on this blog in Valuations and Completions.

A more technical discussion of the Hodge conjecture, Kahler manifolds, and complex algebraic geometry can be found in the following lecture of Claire Voisin, which is part of the Proceedings of the 2010 International Congress of Mathematicians in Hyderabad, India:

On the Cohomology of Algebraic Varieties by Claire Voisin

More about these subjects will hopefully be discussed on this blog at sometime in the future.

# An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry

String theory is by far the most popular of the current proposals to unify the as of now still incompatible theories of quantum mechanics and general relativity. In this post we will give a short overview of the concepts involved in string theory, but not with the goal of discussing the theory itself in depth (hopefully there will be more posts in the future working towards this task). Instead, we will focus on introducing a very interesting and very beautiful branch of mathematics that arose out of string theory called mirror symmetry. In particular, we will focus on a version of it originally formulated by the mathematician Maxim Kontsevich in 1994 called homological mirror symmetry.

We will start with string theory. String theory started out as a theory of the nuclear forces that held together the protons and electrons in the nucleus of an atom. It was abandoned later on, due to a more successful theory called quantum chromodynamics taking its place. However, it was soon found out that string theory could model the elusive graviton, a particle “carrier” of gravity in the same way that a photon is a particle “carrier” of electromagnetism (the photon is more popularly referred to as a particle of light, but because light itself is an electromagnetic wave, it is also a manifestation of an electromagnetic field), and since then physicists have started developing string theory, no longer in the sole context of nuclear forces, but as a possible candidate for a working theory of quantum gravity.

The incompatibility of quantum mechanics and general relativity (which is currently our accepted theory of gravity) arises from the nonrenormalizability of gravity. In calculations in quantum field theory (see Some Basics of Relativistic Quantum Field Theory and Some Basics of (Quantum) Electrodynamics), there appear certain “nonsensical” quantities which are made sense of via a “corrective” procedure called renormalization (not to be confused with some other procedures called “normalization”). While the way that renormalization works is not really completely understood at the moment, it is known that this procedure at least “works” – this means that it produces the correct values of quantities, as can be checked via experiment.

Renormalization, while it works for the other forces, however fails for gravity. Roughly this is sometimes described as gravity “wildly fluctuating” at the smallest scales. What we know is that this signals, for us, a lack of knowledge of  what physics is like at these extremely small scales (much smaller than the current scale of quantum mechanics).

String theory attempts to solve this conundrum by proposing that particles, at the very smallest scales, are not “particles” at all, but “strings”. This takes care of the problem of fluctuations at the smallest scales, since there is a limit to how small the scale can be, set by the length of the strings. It is perhaps worth noting at this point that the next most popular contender to string theory, loop quantum gravity, tackles this problem by postulating that space itself is not continuous, but “discretized” into units of a certain length. For both theories, this length is predicted to be around $10^{-35}$ meters, a constant quantity which is known as the Planck length.

Over time, as string theory was developed, it became more ambitious, aiming to provide not only the unification of quantum mechanics and general relativity, but also the unification of the four fundamental forces – electromagnetism, the weak nuclear force, the strong nuclear force, and gravity, under one “theory of everything“. At the same time, it needed more ingredients – to be able to account for bosons, the particles carrying “forces”, such as photons and gravitons, and the fermions, particles that make up matter, such as electrons, protons, and neutrons, a new ingredient had to be added, called supersymmetry. In addition, it worked not in the four dimensions of spacetime that we are used to, but instead required ten dimensions (for the “bosonic” string theory, before supersymmetry, the number of dimensions required was a staggering twenty-six)!

How do we explain spacetime having ten dimensions, when we experience only four? It turns out, even before string theory, the idea of extra dimensions was already explored by the physicists Theodor Kaluza and Oskar Klein. They proposed a theory unifying electromagnetism and gravity by postulating an “extra” dimension which was “curled up” into a loop so small we could never notice it. The usual analogy is that of an ant crossing a wire – when the radius of the wire is big, the ant realizes that it can go sideways along the wire, but when the radius of the wire is small, it is as if there is only one dimension that the ant can move along.

So we now have this idea of six curled up dimensions of spacetime, in addition to the usual four. It turns out that there are so many ways that these dimensions can be curled up. This phenomenon is called the string theory landscape, and it is one of the biggest problems facing string theory today. What could be the specific “shape” in which these dimensions are curled up, and why are they not curled up in some other way? Some string theorists answer this by resorting to the controversial idea of a multiverse, so that there are actually several existing universes, each with its own way of how the extra six dimensions are curled up, and we just happen to be in this one because, perhaps, this is the only one where the laws of physics (determined by the way the dimensions are curled up) are able to support life. This kind of reasoning is called the anthropic principle.

In addition to the string theory landscape, there was also the problem of having several different versions of string theory. These problems were perhaps alleviated by the discovery of mysterious dualities. For example, there is the so-called T-duality, where a compactification (a “curling up”) with a bigger radius gives the same laws of physics as a compactification with a smaller, “reciprocal” radius. Not only do the concept of dualities connect the different ways in which the extra dimensions are curled up, they also connect the several different versions of string theory! In 1995, the physicist Edward Witten conjectured that this is perhaps because all these different versions of string theory come from a single “mother theory”, which he called “M-theory“.

In 1991, physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes used these dualities to solve a mathematical problem that had occupied mathematicians for decades, that of counting curves on a certain manifold (a manifold is a shape without sharp corners or edges) known as a Calabi-Yau manifold. In the context of Calabi-Yau manifolds, which are some of the shapes in which the extra dimensions of spacetime are postulated to be curled up, these dualities are known as mirror symmetry. With the success of Candelas, de la Ossa, Green, and Parkes, mathematicians would take notice of mirror symmetry and begin to study it as a subject of its own.

Calabi-Yau manifolds are but special cases of Kahler manifolds, which themselves are very interesting mathematical objects because they can be studied using three aspects of differential geometry – Riemannian geometry, symplectic geometry, and complex geometry.

We have already encountered examples of Kahler manifolds on this blog – they are the elliptic curves (see Elliptic Curves and The Moduli Space of Elliptic Curves). In fact elliptic curves are not only Kahler manifolds but also Calabi-Yau manifolds, and they are the only two-dimensional Calabi-Yau manifolds (we sometimes refer to them as “one-dimensional” when we are considering “complex dimensions”, as is common practice in algebraic geometry – this apparent “discrepancy” in counting dimensions arises because we need two real numbers to specify a complex number). In string theory of course we consider six-dimensional (three-dimensional when considering complex dimensions) Calabi-Yau manifolds, since there are six extra curled up dimensions of spacetime, but often it is also fruitful to study also the other cases, especially the simpler ones, since they can serve as our guide for the study of the more complicated cases.

Riemannian geometry studies Riemannian manifolds, which are manifolds equipped with a metric tensor, which intuitively corresponds to an “infinitesimal distance formula” dependent on where we are on the manifold. We have already encountered Riemannian geometry before in Geometry on Curved Spaces and Connection and Curvature in Riemannian Geometry. There we have seen that Riemannian geometry is very important in the mathematical formulation of general relativity, since in this theory gravity is just the curvature of spacetime, and the metric tensor expresses this curvature by showing how the formula for the infinitesimal distance between two points (actually the infinitesimal spacetime interval between two events) changes as we move around the manifold.

Symplectic geometry, meanwhile, studies symplectic manifolds. If Riemannian manifolds are equipped with a metric tensor that measures “distances”, symplectic manifolds are equipped with a symplectic form that measures “areas”. The origins of symplectic geometry are actually related to William Rowan Hamilton’s formulation of classical mechanics (see Lagrangians and Hamiltonians), as developed later on by Henri Poincare. There the object of study is phase space, which gives the state of a system based on the position and momentum of the objects that comprise it. It is this phase space that is expressed as a symplectic manifold.

Complex geometry, following our pattern, studies complex manifolds. These are manifolds which locally look like $\mathbb{C}^{n}$, in the same way that ordinary differentiable manifolds locally look like $\mathbb{R}^{n}$. Just as Riemannian geometry has metric tensors and symplectic geometry has symplectic forms, complex geometry has complex structures, mappings of tangent spaces with the property that applying them twice is the same as multiplication by $-1$, mimicking the usual multiplication by the imaginary unit $i$ on the complex plane.

Complex manifolds are not only part of differential geometry, they are also often studied using the methods of algebraic geometry! We recall (see Basics of Algebraic Geometry) that algebraic geometry studies varieties and schemes, which are shapes such as lines, conic sections (parabolas, hyperbolas, ellipses, and circles), and elliptic curves, that can be described by polynomials (their modern definitions are generalizations of this concept). In fact, all Calabi-Yau manifolds can be described by polynomials, such as the following example, due to user Andrew J. Hanson of Wikipedia:

This is a visualization (actually a sort of “cross section”, since we can only display two dimensions and this object is actually six-dimensional) of the Calabi-Yau manifold described by the following polynomial equation:

$\displaystyle V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0$

This polynomial equation (known as the Fermat quintic) actually describes the Calabi-Yau manifold  in projective space using homogeneous coordinates. This means that we are using the concepts of projective geometry (see Projective Geometry) to include “points at infinity“.

We note at this point that Kahler manifolds and Calabi-Yau manifolds are interesting in their own right, even outside of the context of string theory. For instance, we have briefly mentioned in Algebraic Cycles and Intersection Theory the Hodge conjecture, one of seven “Millenium Problems” for which the Clay Mathematics Institute is currently offering a million-dollar prize, and it concerns Kahler manifolds. Perhaps most importantly, it “unifies” several different branches of mathematics; as we have already seen, the study of Kahler manifolds and Calabi-Yau manifolds involves Riemannian geometry, symplectic geometry, complex geometry, and algebraic geometry. The more recent version of mirror symmetry called homological mirror symmetry further adds category theory and homological algebra to the mix.

Now what mirror symmetry more specifically states is that a version of string theory called Type IIA string theory, on a spacetime with extra dimensions compactified onto a certain Calabi-Yau manifold $V$, is the same as another version of string theory, called Type IIB string theory, on a spacetime with extra dimensions compactified onto another Calabi-Yau manifold $W$, which is “mirror” to the Calabi-Yau manifold $V$.

The statement of homological mirror symmetry (which is still conjectural, but mathematically proven in certain special cases) expresses the idea of the previous paragraph as follows (quoted verbatim from the paper Homological Algebra of Mirror Symmetry by Maxim Kontsevich):

Let $(V,\omega)$ be a $2n$-dimensional symplectic manifold with $c_{1}(V)=0$ and $W$ be a dual $n$-dimensional complex algebraic manifold.

The derived category constructed from the Fukaya category $F(V)$ (or a suitably enlarged one) is equivalent to the derived category of coherent sheaves on a complex algebraic variety $W$.

The statement makes use of the language of category theory and homological algebra (see Category TheoryMore Category Theory: The Grothendieck ToposEven More Category Theory: The Elementary ToposExact SequencesMore on Chain Complexes, and The Hom and Tensor Functors), but the idea that it basically expresses is that there exists a relation between the symplectic aspects of the Calabi-Yau manifold $V$, as encoded in its Fukaya category, and the complex aspects of the Calabi-Yau manifold $W$, as encoded in its category of coherent sheaves (see Sheaves and More on Sheaves). As we have said earlier, the subjects of algebraic geometry and complex geometry are closely related, and hence the language of sheaves show up in (and is an important part of) both subjects. The concept of derived categories, which generalize derived functors like the Ext and Tor functors, allow us to relate the two categories, which otherwise would be expressing different concepts. Inspired by string theory, therefore, we have now a deep and beautiful idea in geometry, relating its different aspects.

Is string theory the correct way towards a complete theory of quantum gravity, or the so-called “theory of everything”? As of the moment, we don’t know. Quantum gravity is a very difficult problem, and the scales involved are still far out of our reach – in order to probe smaller and smaller scales we need particle accelerators with higher and higher energies, and right now the technologies that we have are still very, very far from the scales which are relevant to quantum gravity. Still, it is hoped for that whatever we find in experiments in the near future, not only in the particle accelerators but also in the radio telescopes that look out into space, will at least guide us towards the correct path.

There are some who believe that, in the absence of definitive experimental evidence, mathematical beauty is our next best guide. And, without a doubt, string theory is related to, and has inspired, some very beautiful and very interesting mathematics, including that which we have discussed in this post. Still, physics, like all natural science, is empirical (based on evidence and observation), and hence it is ultimately physical evidence that will be the judge of correctness. It may yet turn out that string theory is wrong, and that it is a different theory which describes the fundamental physical laws of nature, or that it needs drastic modifications to its ideas. This will not invalidate the mathematics that we have described here, anymore than the discoveries of Copernicus invalidated the mathematics behind the astronomical model of Ptolemy – in fact this mathematics not only outlived the astronomy of Ptolemy, but served the theories of Copernicus, and his successors, just as well. Hence we cannot really say that the efforts of Ptolemy were wasted, since even though his scientific ideas were shown to be wrong, still his mathematical methods were found very useful by those who succeeded him. Thus, while our current technological limitations prohibit us from confirming or ruling out proposals for a theory of quantum gravity such as string theory, there is still much to be gained from such continued efforts on the part of theory, while experiment is still in the process of catching up.

Our search for truth continues. Meanwhile, we have beauty to cultivate.

References:

String Theory on Wikipedia

Mirror Symmetry on Wikipedia

Homological Mirror Symmetry on Wikipedia

Calabi-Yau Manifold on Wikipedia

Kahler Manifold on Wikipedia

Riemannian Geometry on Wikipedia

Symplectic Geometry on Wikipedia

Complex Geometry on Wikipedia

Fukaya Category on Wikipedia

Coherent Sheaf on Wikipedia

Derived Category on Wikipedia

Image by User Andrew J. Hanson of Wikipedia

Homological Algebra of Mirror Symmetry by Maxim Kontsevich

The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory by Brian Greene

String Theory by Joseph Polchinski

String Theory and M-Theory: A Modern Introduction by Katrin Becker, Melanie Becker, and John Schwarz

# Algebraic Cycles and Intersection Theory

In this post, we will take on intersection theory – which is pretty much just what it sounds like, except for a few modifications which we will later discuss. For example, we may ask, where do the curves $y=x^{2}$ (a parabola) and the line $y=1$ (a horizontal line) intersect? It only takes a little high school algebra and analytic geometry (which is really a more elementary form of what we now more properly call algebraic geometry) to find that they intersect at the two points $(-1,1)$ and $(1,1)$.

Suppose, instead, that we want to take the intersection of the parabola $y=x^{2}$ and the horizontal line $y=-1$. If the coordinates $x$ and $y$ are real numbers, we would have no intersection. But if they are complex numbers, then we will find that they do intersect, once again at two points, namely $(-i,-1)$ and $(i,-1)$. When complex numbers are involved, it may be difficult to visualize things – for example, the complex numbers are often visualized as a plane, not as a line – but we will continue to refer, say, to $y=-1$ as a “line”. This is rather common practice in algebraic geometry – recall that we have also been referring to a torus as an elliptic curve (see The Moduli Space of Elliptic Curves)!

Consider now the intersection of the parabola $y=x^{2}$ and the horizontal line $y=0$. This time, contrary to the earlier two cases, the curves intersect only at one point, namely at $(0,0)$. But we would like to think of them as intersecting “twice”, even though the intersection occurs only at a single point. Hence, we say that the point $(0,0)$ has intersection multiplicity equal to $2$.

The notion of intersection multiplicities make sense – generally speaking, for instance, imagining a random parabola and a random line in the $x$$y$ plane, they will generally intersect at two points, except in certain instances, such as when the line is tangent to the parabola – this is of course the special, less general, case. In order to make our counting of intersections consistent, even with these special cases, we need this idea of “intersection multiplicities.”

Another way to think of the previous example is that the parabola and the line having only one intersection is such a special case that simply “displacing” or “moving” either curve by a little bit results in them having two intersections. Consider, for example, the following diagram courtesy of user Jakob.scholbach of Wikipedia:

Equipped with the idea of intersection multiplicities (which we will explicitly give the formula for later), we have Bezout’s theorem, which states that the number of intersections of two curves, counted with their intersection multiplicities, is equal to the product of the degrees of the polynomials that define them. For example, two parabolas will generally intersect at four points, except in special cases where their intersections have multiplicities greater than $1$.

For higher-dimensional varieties, the intersections need not be points, but other kinds of varieties. For an $n$-dimensional variety $W$ embedded in a larger $m$-dimensional variety $V$ (which we may think of as the space the $n$-dimensional variety is living in), the codimension of $W$ in $V$ is given by $m-n$. If the codimension of the intersection of two varieties is equal to the sum of the codimensions of the intersecting varieties, then we say that the intersection is proper.

Proper intersections correspond to our intuition. For example, consider again curves such as parabolas and lines in the plane. The plane is $2$-dimensional, while curves are $1$ dimensional. Therefore their codimension in the plane is equal to $2-1=1$. Proper intersections will then be points, which have dimension equal to $0$ and therefore have codimension in the plane equal to $2$. Similarly, the proper intersection of two surfaces, for example two planes, in some $3$-dimensional space, is a curve (a line in the case of two planes), since surfaces have codimension equal to $1$ inside the $3$-dimensional space, while curves have codimension equal to $2$.

We can now give the definition of the intersection multiplicity. It is quite technical, involving the Tor functor (see The Hom and Tensor Functors), but we will also give the special case for curves, which is a little less technical compared to the general case. Let $V$ and $W$ be two subvarieties of some smooth variety $X$ (for a discussion of smoothness and singularities see Reduction of Elliptic Curves Modulo Primes) which intersect properly and let $Z$ be their set-theoretic intersection. Then the intersection multiplicity $\mu(Z;V,W)$ is given by

$\displaystyle \mu(Z;V,W)=\sum_{i=0}^{\infty}(-1)^{i}\text{length}_{\mathcal{O}_{X,z}}(\text{Tor}_{\mathcal{O}_{X,z}}^{i}(\mathcal{O}_{X,z}/I, \mathcal{O}_{X,z}/J))$

where $I$ and $J$ are the ideals corresponding to the varieties $V$ and $W$ respectively, and $z$ is the generic point of the variety $Z$ (we are using here the definition of a variety as a scheme satisfying certain conditions, which we have not actually discussed in this blog yet – we will leave this to the references for now). The concept of length is a generalization of the concept of dimension in algebraic geometry, and refers to the length (the ordinary use of the term) of the longest chain of modules that contain one another (while dimension refers to the length of the longest chain of rings that contain one another).

If $V$ and $W$ are curves on a surface $X$ then the above formula reduces to

$\displaystyle \mu(Z;V,W)=\text{length}_{\mathcal{O}_{X,z}}(\mathcal{O}_{X,z}/I\otimes_{\mathcal{O}_{X,z}} \mathcal{O}_{X,z}/J)$.

Another concept in algebraic geometry closely related to intersection theory is that of an algebraic cycle. Algebraic cycles generalize the idea of divisors (see Divisors and the Picard Group). Algebraic cycles on a variety $X$ can be thought of as “linear combinations” of the subvarieties (satisfying certain conditions, such as being closed, reduced, and irreducible, so that they are not unions of other subvarieties) on $X$. Divisors themselves are just algebraic cycles of codimension $1$; in other words, they are algebraic cycles whose dimension is $1$ less than the variety in which they are embedded. In Divisors and the Picard Group, we considered curves, which are varieties of dimension $1$, hence the divisors on the curves were linear combinations (with integer coefficients) of points, i.e. subvarieties of dimension equal to $0$.

Analogous to the Picard group for divisors we have the Chow group of algebraic cycles modulo rational equivalence. Two algebraic cycles $V$ and $W$ on a variety $Y$ are said to be rationally equivalent if there is a rational function $f:Y\rightarrow \mathbb{P}$ such that $V-W=f^{-1}(0)-f^{-1}(\infty)$, counting multiplicities. Chow’s moving lemma states that for any two algebraic cycles $V$ and $W$ on a smooth, quasi-projective (quasi-projective means it is the intersection of a Zariski-open and Zariski-closed subset in some projective space) variety $X$, there exists another algebraic cycle $W'$ rationally equivalent to $W$ such that $V$ and $W$ intersect properly. Besides rational equivalence, there are also other notions of equivalence for algebraic cycles, such as algebraic, homological, and numerical equivalence; all of these are important in the study of algebraic cycles, but we will leave them to the references for now.

Taking intersections of subvarieties gives the Chow group a ring structure (we therefore have the concept of an intersection product). In this context we may also refer to the Chow group as the Chow ring. The Chow ring is also an example of a graded ring, which means that the intersection product is a mapping that sends a pair of equivalence classes of algebraic cycles, one with codimension $i$ and another with codimension $j$, to an equivalence class of algebraic cycles with codimension $i+j$:

$\displaystyle CH^{i}\times CH^{j}\rightarrow CH^{i+j}$.

Algebraic cycles on a smooth variety are related to cohomology (see Homology and Cohomology and Cohomology in Algebraic Geometry) via the notion of a cycle map:

$\displaystyle \text{cl}:CH^{j}(X)\rightarrow H^{2j}(X)$.

The intersection product carries over into cohomology, corresponding to the so-called cup product of cohomology classes. Actually, there are many cohomology theories, but the ones considered to be “good” cohomology theories (more technically, they are the ones referred to as the Weil cohomology theories) are required to have a cycle map. Related to the notion of the cycle map is the famous Hodge conjecture in complex algebraic geometry, which states that under a certain well-known decomposition of the cohomology groups $H^{k}=\oplus_{p+q=k}H^{p,q}$, all cohomology classes of a certain kind (the so-called Hodge classes) come from algebraic cycles. Another similar conjecture is the Tate conjecture, which relates the cohomology classes coming from algebraic cycles to the elements that are fixed by the action of the Galois group (see Galois Groups). Other important conjectures in the study of algebraic cycles are the so-called standard conjectures formulated by Alexander Grothendieck as part of his strategy to prove the Weil conjectures (see The Riemann Hypothesis for Curves over Finite Fields). The Weil conjectures were proved without the need to prove the standard conjectures, but the standard conjectures themselves continue to be the object of modern mathematical research.

References:

Intersection Theory on Wikipedia

Bezout’s Theorem on Wikipedia

Algebraic Cycle on Wikipedia

Chow group on Wikipedia

Chow’s Moving Lemma on Wikipedia

Motives – Grothendieck’s Dream by James S. Milne

The Riemann Hypothesis over Finite Fields: From Weil to the Present Day by J. S. Milne

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne