### algebraic geometry

The purpose of this post, the third in a series on deformation theory and DGLAs, is to describe the obstruction theory for a formal moduli problem associated to a DGLA.

1. Tangent-obstruction theories

Standard problems in classical deformation theory usually have a “tangent-obstruction theory” parametrized by certain successive cohomology groups. For example, let’s consider the problem of deformations of a smooth variety ${X}$ over an algebraically closed field ${k}$, over finite-dimensional local ${k}$-algebras. Then:

• The “infinitesimal automorphisms” of ${X}$—that is, automorphisms of the trivial deformation over ${k[\epsilon]/\epsilon^2}$—are given by ${H^0( X, T_X)}$ where ${T_X}$ is the tangent bundle (i.e., vector fields).
• The isomorphism classes of deformations of ${X}$ over the dual numbers ${k[\epsilon]/\epsilon^2}$ are given by ${H^1(X, T_X)}$.
• There is an obstruction theory with ${H^1, H^2}$. Specifically, given a square-zero extension of finite-dimensional local ${k}$-algebras

$\displaystyle 0 \rightarrow I \rightarrow A' \rightarrow A \rightarrow 0,$

and given a deformation ${\xi}$ of ${X}$ over ${\mathrm{Spec} A}$, there is a functorial obstruction in ${H^2(X, T_X) \otimes_k I}$ to extending the deformation over the inclusion ${\mathrm{Spec} A \hookrightarrow \mathrm{Spec} A'}$.

• In the previous item, if the obstruction vanishes, then the isomorphism classes of extensions of ${\xi}$ over ${\mathrm{Spec} A'}$ are a torsor for ${H^1(X, T_X) \otimes_k I}$.

One has a similar picture for other deformation problems, for example deformations of vector bundles or closed subschemes. The “derived” approach to deformation theory provides (at least in characteristic zero) a general explanation for this phenomenon. (more…)

Let ${k}$ be a field of characteristic zero. In the previous post, we introduced the category (i.e., ${\infty}$-category) ${\mathrm{Moduli}_k}$ of formal moduli problems over ${k}$. A formal moduli problem over ${k}$ is a moduli problem, taking values in spaces, that can be evaluated on the class of “derived” artinian ${k}$-algebras with residue field ${k}$: this was the category ${\mathrm{CAlg}_{sm}}$ introduced in the previous post.

In other words, a formal moduli problem was a functor

$\displaystyle F: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S} \ (= \text{spaces}),$

which was required to send ${k}$ itself to a point, and satisfy a certain cohesiveness condition: ${F}$ respects certain pullbacks in ${\mathrm{CAlg}_{sm}}$ (which corresponded geometrically to pushouts of schemes).

The main goal of the series of posts was to sketch a proof of (and define everything in) the following result:

Theorem 7 (Lurie; Pridham) There is an equivalence of categories between ${\mathrm{Moduli}_k}$ and the ${\infty}$-category ${\mathrm{dgLie}}$ of DGLAs over ${k}$.

4. Overview

Here’s a rough sketch of the idea. Given a formal moduli problem ${F}$, we should think of ${F}$ as something like a small space, concentrated at a point but with lots of “infinitesimal” thickening. (Something like a ${\mathrm{Spf}}$.) Moreover, ${F}$ has a canonical basepoint corresponding to the “trivial deformation.” That is, we can think of ${F}$ as taking values in pointed spaces rather than spaces.

It follows that we can form the loop space ${\Omega F = \ast \times_F \ast}$ of ${F}$, which is a new formal moduli problem. However, ${\Omega F}$ has more structure: it’s a group object in the category of formal moduli problems — that is, it’s some sort of derived formal Lie group. Moreover, knowledge of the original ${F}$ is equivalent to knowledge of ${\Omega F}$ together with its group structure: we can recover ${F}$ as ${B \Omega F}$ (modulo connectivity issues that end up not being a problem). This relation between ordinary objects and group objects (via ${B, \Omega}$) is something very specific to the derived or homotopy world, and it’s what leads to phenomena such as Koszul duality. (more…)

There’s a “philosophy” in deformation theory that deformation problems in characteristic zero come from dg-Lie algebras. I’ve been trying to learn a little about this. Precise statements have been given by Lurie and Pridham which consider categories of “derived” deformation problems (i.e., deformation problems that can be evaluated on derived rings) and establish equivalences between them and suitable (higher) categories of dg-Lie algebras. I’ve been reading in particular Lurie’s very enjoyable survey of his approach to the problem, which sketches the equivalence in an abstract categorical context with the essential input arising from Koszul duality between Lie algebras and commutative algebras. In this post, I’d just like to say what a “deformation problem” is in the derived world.

1. Introduction

Let ${\mathcal{M}}$ be a classical moduli problem. Abstractly, we will think of ${\mathcal{M}}$ as a functor

$\displaystyle \mathcal{M}:\mathrm{Ring} \rightarrow \mathrm{Sets},$

such that, for a (commutative) ring ${R}$, the set ${\mathcal{M}(R)}$ will be realized as maps from ${\mathrm{Spec} R}$ into a geometric object—a scheme or maybe an algebraic space.

Example 1${\mathcal{M}}$ could be the functor that sends ${R}$ to the set of closed subschemes of ${\mathbb{P}^n_R}$ which are flat over ${R}$. In this case, ${\mathcal{M}}$ comes from a scheme: the Hilbert scheme.

We want to think of ${\mathcal{M}}$ as some kind of geometric object and, given a point ${x: \mathrm{Spec} k \rightarrow \mathcal{M}}$ for ${k}$ a field (that is, an element of ${\mathcal{M}(k)}$), we’d like to study the local structure of ${\mathcal{M}}$ near ${x}$. (more…)

Let ${X}$ be an abelian variety over the algebraically closed field ${k}$. In the previous post, we studied the Picard scheme ${\mathrm{Pic}_X}$, or rather its connected component ${\mathrm{Pic}^0_X}$ at the identity. The main result was that ${\mathrm{Pic}^0_X}$ was itself an abelian variety (in particular, smooth) of the same dimension as ${X}$, which parametrizes precisely the translation-invariant line bundles on ${X}$.

We also saw how to construct isogenies between ${X}$ and ${\mathrm{Pic}^0_X}$. Given an ample line bundle ${\mathcal{L}}$ on ${X}$, the map

$\displaystyle X \rightarrow \mathrm{Pic}^0_X, \quad x \mapsto t_x^* \mathcal{L} \otimes \mathcal{L}^{-1}$

is an isogeny. Such maps were in fact the basic tool in proving the above result.

The goal of this post is to show that the contravariant functor

$\displaystyle X \mapsto \mathrm{Pic}^0_X$

from abelian varieties over ${k}$ to abelian varieties over ${k}$, is a well-behaved duality theory. In particular, any abelian variety is canonically isomorphic to its bidual. (This explains why the double Picard functor on a general variety is the universal abelian variety generated by that variety, the so-called Albanese variety.) In fact, we won’t quite finish the proof in this post, but we will finish the most important step: the computation of the cohomology of the universal line bundle on $X \times \mathrm{Pic}^0_X$.

Motivated by this, we set the notation:

Definition 11 We write ${\hat{X}}$ for ${\mathrm{Pic}^0_X}$.

The main reference for this post is Mumford’s Abelian varieties. (more…)

Let ${k}$ be an algebraically closed field, and ${X}$ a projective variety over ${k}$. In the previous two posts, we’ve defined the Picard scheme ${\mathrm{Pic}_X}$, stated (without proof) the theorem of Grothendieck giving conditions under which it exists, and discussed the infinitesimal structure of ${\mathrm{Pic}_X}$ (or equivalently of the connected component ${\mathrm{Pic}^0_X}$ at the origin).

We saw in particular that the tangent space to the Picard scheme could be computed via

$\displaystyle T \mathrm{Pic}^0_X = H^1(X, \mathcal{O}_X),$

by studying deformations of a line bundle over the dual numbers. In particular, in characteristic zero, a simply connected smooth variety has trivial ${\mathrm{Pic}^0_X}$. To get interesting ${\mathrm{Pic}_X^0}$‘s, we should be looking for non-simply connected varieties: abelian varieties are a natural example.

Let ${X}$ be an abelian variety over ${k}$. The goal in this post is to describe ${\mathrm{Pic}^0_X}$, which we’ll call the dual abelian variety (we’ll see that it is in fact smooth). We’ll in particular identify the line bundles that it parametrizes. Most of this material is from David Mumford’s Abelian varieties and Alexander Polischuk’s Abelian varieties, theta functions, and the Fourier transform. I also learned some of it from a class that Xinwen Zhu taught last spring; my (fairly incomplete) notes from that class are here(more…)

Let ${X}$ be a projective variety over the algebraically closed field ${k}$, endowed with a basepoint ${\ast}$. In the previous post, we saw how to define the Picard scheme ${\mathrm{Pic}_X}$ of ${X}$: a map from a ${k}$-scheme ${Y}$ into ${\mathrm{Pic}_X}$ is the same thing as a line bundle on ${Y \times_k X}$ together with a trivialization on ${Y \times \ast}$. Equivalently, ${\mathrm{Pic}_X}$ is the sheafification (in the Zariski topology, even) of the functor

$\displaystyle Y \mapsto \mathrm{Pic}(X \times_k Y)/\mathrm{Pic}(Y),$

so we could have defined the functor without a basepoint.

We’d like to understand the local structure of ${\mathrm{Pic}_X}$ (or, equivalently, of ${\mathrm{Pic}^0_X}$), and, as with moduli schemes in general, deformation theory is a basic tool. For example, we’d like to understand the tangent space to ${\mathrm{Pic}_X}$ at the origin ${0 \in \mathrm{Pic}_X}$. The tangent space (this works for any scheme) can be identified with

$\displaystyle \hom_{0}( \mathrm{Spec} k[\epsilon]/\epsilon^2, \mathrm{Pic}_X).$ (more…)

Let ${X}$ be an abelian variety over an algebraically closed field ${k}$. If ${k = \mathbb{C}}$, then ${X}$ corresponds to a complex torus: that is, ${X}$ can be expressed complex analytically as ${V/\Lambda}$ where ${V}$ is a complex vector space of dimension ${\dim X}$ and ${\Lambda \subset V}$ is a lattice (i.e., a ${\mathbb{Z}}$-free, discrete submodule of rank ${2g}$). In this case, one can form the dual abelian variety

$\displaystyle X^{\vee} = \hom(X, S^1) = \hom_{\mathrm{cont}}(V/\Lambda, \mathbb{R}/\mathbb{Z}) \simeq \hom_{\mathbb{R}}(V, \mathbb{R})/2\pi i \hom(\Lambda, \mathbb{Z}).$

At least, ${X^{\vee}}$ as defined is a complex torus, but it turns out to admit the structure of an abelian variety.

The purpose of the next few posts is to describe an algebraic version of this duality: it turns out that ${X^{\vee}}$ can be constructed as a scheme, purely algebraically. I’d like to start with a couple of posts on Picard schemes. A useful reference here is this article of Kleiman.

1. The Picard scheme analytically

Let ${X }$ be a smooth projective variety over the complex numbers ${\mathbb{C}}$. The collection of line bundles ${\mathrm{Pic}(X)}$ is a very interesting invariant of ${X}$. Usually, it splits into two pieces: the “topological” piece and the “analytic” piece. For instance, there is a first Chern class map

$\displaystyle c_1: \mathrm{Pic}(X) \rightarrow H^2(X; \mathbb{Z}) ,$

which picks out the topological type of a line bundle. (Topologically, line bundles on a space are classified by their first Chern class.) The admissible topological types are precisely the classes in ${H^2(X; \mathbb{Z})}$ which project to ${(1,1)}$-classes in ${H^2(X; \mathbb{C})}$ under the Hodge decomposition. (more…)

Let ${X}$ be a variety over an algebraically closed field ${k}$. ${X}$ is said to be rational if ${X}$ is birational to ${\mathbb{P}_k^n}$. In general, it is difficult to determine when a variety in higher dimensions is rational, although there are numerical invariants in dimensions one and two.

• Let ${X}$ be a smooth projective curve. Then ${X}$ is rational if and only if its genus is zero.
• Let ${X}$ be a smooth projective surface. Then ${X}$ is rational if and only if there are no global 1-forms on ${X}$ (i.e., ${H^0(X, \Omega_{X/k}) = 0}$) and the second plurigenus ${H^0(X, \omega_{X/k}^{\otimes 2}) }$ vanishes. This is a statement about the negativity of the cotangent bundle (or, equivalently, of the positivity of the tangent bundle) which is a birational invariant and which holds for ${\mathbb{P}^2_k}$. The result is a criterion of Castelnuovo, extended by Zariski to characteristic $p$.

In higher dimensions, it is harder to tell when a variety is rational. An easier problem is to determine when a variety is unirational: that is, when there is a dominant rational map

$\displaystyle \mathbb{P}_k^n \dashrightarrow X;$

or, equivalently, when the function field ${k(X)}$ has a finite extension which is purely transcendental. In dimensions one and two (and in characteristic zero), the above invariants imply that a unirational variety is rational. In higher dimensions, there are many more unirational varieties: for example, a theorem of Harris, Mazur, and Pandharipande states that a degree ${d}$ hypersurface in ${\mathbb{P}^N}$, ${N \gg 0}$ is always unirational.

The purpose of this post is to describe a theorem of Serre that shows the difficulty of distinguishing rationality from unirationality. Let’s work over ${\mathbb{C}}$. The fundamental group of a smooth projective variety is a birational invariant, and so any rational variety has trivial ${\pi_1}$.

Theorem 1 (Serre) A unirational (smooth, projective) variety over ${\mathbb{C}}$ has trivial ${\pi_1}$.

The reference is Serre’s paper “On the fundamental group of a unirational variety,” in J. London Math Soc. 1959. (more…)

Let ${S}$ be a smooth, projective surface over the algebraically closed field ${k}$. Previous posts have set up an intersection theory

$\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z}$

on ${S}$ with very convenient formal properties. We also described a historically important use of this machinery: the Weil bound on points on a smooth curve over a finite field. The purpose of this post is to prove an entirely numerical criterion for ampleness of a line bundle on a surface, due to Nakai and Moishezon.

Let ${D}$ be a very ample divisor on ${S}$. Then we have:

• ${D.C > 0}$ for all curves (i.e., strictly effective divisors) on ${S}$. In fact, if ${D}$ defines an imbedding ${S \hookrightarrow \mathbb{P}^M}$, then the degree of ${C}$ under this imbedding is ${D.C}$.
• As a special case of this, ${D.D > 0}$. In fact, ${D}$ must be effective.

Since a power of an ample divisor is very ample, the same is true for an ample divisor.

The purpose of this post is to prove the converse:

Theorem 1 (Nakai-Moishezon) Let ${S}$ be a smooth projective surface as above. If ${D}$ is a divisor on ${S}$ (not necessarily effective!) satisfying ${D.D>0}$ and ${D.C > 0}$ for all curves on ${S}$, then ${D}$ is ample. In particular, ampleness depends only on the numerical equivalence class of ${D}$.

Once again, the source for this material is Hartshorne’s Algebraic geometry. The goal is to get to some computations and examples as soon as possible.   (more…)

The purpose of this post is to describe an application of the general intersection theory machinery (for curves on surfaces) developed in the previous posts: the Weil bound on points on a curve over a finite field.

1. Statement of the Weil bound

Let ${C}$ be a smooth, projective, geometrically irreducible curve over ${\mathbb{F}_q}$ of genus $g$. Then the Weil bound states that:

$\displaystyle |C(\mathbb{F}_q) - q - 1 | \leq 2 g \sqrt{q}.$

Weil’s proof of this bound is based on intersection theory on the surface ${C \times C}$. More precisely, let

$\displaystyle \overline{C} = C \times_{\mathbb{F}_q} \overline{\mathbb{F}_q},$

so that ${\overline{C}}$ is a smooth, connected, projective curve. It comes with a Frobenius map

$\displaystyle F: \overline{C} \rightarrow \overline{C}$

of ${\overline{\mathbb{F}_q}}$-varieties: in projective coordinates the Frobenius runs

$\displaystyle [x_0: \dots : x_n] \mapsto [x_0^q: \dots : x_n^q].$

In particular, the map has degree ${q}$. One has

$\displaystyle C( \mathbb{F}_q) = \mathrm{Fix}(F, \overline{C}(\overline{\mathbb{F}}_q))$

representing the ${\mathbb{F}_q}$-valued points of ${C}$ as the fixed points of the Frobenius (Galois) action on the ${\overline{\mathbb{F}_q}}$-valued points. So the strategy is to count fixed points, using intersection theory.

Using the (later) theory of ${l}$-adic cohomology, one represents the number of fixed points of the Frobenius as the Lefschetz number of ${F}$: the action of ${F}$ on ${H^0}$ and ${H^2}$ give the terms ${q+1}$. The fact that (remaining) action of ${F}$ on the ${2g}$-dimensional vector space ${H^1}$ can be bounded is one of the Weil conjectures, proved by Deligne for general varieties: here it states that ${F}$ has eigenvalues which are algebraic integers all of whose conjugates have absolute value ${\sqrt{q}}$. (more…)

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