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…)

It’s been a busy semester, and I haven’t done a great job of updating this blog lately. I have a couple of posts in preparation, but in the meantime:
• I gave a talk on the nilpotence and periodicity theorems in stable homotopy theory at the pre-Talbot seminar (a.k.a. Juvitop) at MIT. All the talks this semester were videotaped; the video of mine is here. The results are really beautiful, showing that the “global” picture of stable homotopy theory exactly parallels the geometry of the moduli stack of formal groups.
• I’ve been taking notes from a course of Joe Harris on the representation theory of Lie groups. Unfortunately, I’m unable to include the many pictures that were drawn in lectures, and the notes are somewhat incomplete.
• I’m spending the summer at the REU program at Emory, and I’ll be thinking about problems in moduli of curves. It should be interesting to get a little experience with algebraic geometry. In particular, I’m going to try to focus this blog in that direction over the next couple of months.

I wrote this for a guest post on Cathy O’Neil’s blog mathbabe.

Climate change is one of those issues that I heard about as a kid, and I assumed naturally that scientists, political leaders, and the rest of the world would work together to solve it. Then I grew up and realized that never happened.

Carbon dioxide emissions are continuing to rise and extreme weather is becoming normal. Meanwhile, nobody in politics seems to want to act, even when major scientific organizations — and now the World Bank — have warned us in the strongest possible terms that the current path towards ${4^{\circ} C}$ or more warming is an absolutely terrible idea (the World Bank called it “devastating”).

A little frustrated, I decided to show up last fall at my school’s umbrella environmental group to hear about the various programs. Intrigued by a curious-sounding divestment campaign, I decided to show up at the first meeting. I had zero knowledge of or experience with the climate movement, and did not realize what it was going to become.

Divestment from fossil fuel companies is a simple and brilliant idea, popularized by Bill McKibben’s article “Global Warming’s Terrifying New Math.” As McKibben observes, there are numerous reasons to divest, both ethical and economic. The fossil fuel reserves of these companies — a determinant of their market value — are five(!) times what scientists estimate can be burned to stay within 2 degree warming. Investing in fossil fuels is therefore a way of betting on climate change. It’s especially absurd for universities to invest in them, when much of the research on climate change took place there.

The other side of divestment is symbolic. It’s not likely that Congress will be able to pass a cap-and-trade or carbon tax system anytime soon, especially when fossil fuel companies are among the biggest contributors to political campaigns. A series of university divestments would draw attention to the problem. It would send a message to the world: that fossil fuel companies should be shunned, for basing their business model on climate change and then for lying about its dangers. This reason echoes the apartheid divestment campaigns of the 1980s.

With support from McKibben’s organization 350.org, divestment took off last fall to become a real student movement, and today, over 300 American universities have active
divestment campaigns from their students. Four universities — Unity College,
Hampshire College, Sterling College, and College of the Atlantic — have already divested. Divestment is spreading both to Canadian universities and to other non-profit organizations. We’ve been covered in the New York Times, endorsed by Al Gore, and, on the other hand, recently featured in a couple of rants by Fox News. (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…)