A Glimpse of Deformation Theory - Lecture Notes | AMS 130, Study notes of United States History

Download A Glimpse of Deformation Theory - Lecture Notes | AMS 130 and more Study notes United States History in PDF only on Docsity! A GLIMPSE OF DEFORMATION THEORY BRIAN OSSERMAN The purpose of these notes, as suggested by the title, is not to provide any sort of comprehensive introduction to deformation theory. Rather, we attempt to convey the main ideas of the theory, with a survey of some applications. We do not explore an exhaustive list of possible topics, nor do we go into details in many proofs. However, in the case of deformations of smooth varieties we do attempt to give a thorough treatment of the theory of first-order infinitesimal deformations, with some hints as to how to generalize to other problems. Because the systematic use of rings with nilpotents is one of the major distin- guishing characteristics of scheme theory, we can view deformation theory as a sub- stantial application of scheme theory that is beyond the reach of classical algrebraic geometry. We hope to provide a relatively accessible and motivated introduction to the theory of cohomology of sheaves, in the form of the Cech cohomology arising in deformations of smooth varieties. 1. The many uses of Artin rings Recall that A is an Artin ring if it is Noetherian of dimension 0. For the sake of brevity, we will in these notes also assume that all Artin rings are local. Recall that the maximal ideal of such a ring consists entirely of nilpotents. Although Spec A for A an Artin ring consists of only a single point, the basic idea motivating deformation theory is that such schemes are still “big enough” to provide useful information. We have already seen Artin rings arising in two different contexts: in tangent spaces, and in criteria for smoothness. In the first case, we had that the tangent space of a scheme X at a point x was equal to the set of maps Spec k[ǫ]/ǫ2 → X with image x, where k = k(x). A special case of the smoothness criterion was that if X is of finite type over a field k, then X is smooth over k if and only if for every Artin k-algebra A, and every quotient A ։ A′ with kernel I a square-zero ideal, every map Spec A′ → X over k can be lifted to a map Spec A → X. Both of these involve understanding maps Spec A → X for A an Artin ring. Now suppose that X is a moduli space representing some moduli functor, or more generally a non-representable moduli functor F̃ . Then the maps Spec A → X (more generally, F̃ (Spec A), which we abbreviate by F̃ (A)) correspond to families of objects over Spec A. If we fix a map Spec k → X (or an element η0 ∈ F̃ (k)), and restrict our attention to A with residue field k, we can consider elements of F̃ (A) restricting to η0 under the natural map Spec k →֒ Spec A, and this corresponds to studying families of objects over Spec A which restrict to the fixed object η0 on the reduced point Spec k; such families are called (infinitesimal) deformations of η0 over Spec A; this motivates the terminology. One of the most basic ideas of deformation theory is that we can study the tangent space and smoothness of a moduli space by considering families over Spec A. 1 2 BRIAN OSSERMAN Although completely classifying families of objects over arbitrary schemes T is usually far too complicated, in the case of Artin rings the bases are generally small enough that one can write down explicit descriptions. 2. (Pre)deformation functors The standard setup (introduced by Schlessinger in [4]) is as follows: we denote the category of Artin rings with a given residue field k by Art(k), and consider (covariant) functors F : Art(k) → Set with the property that F (k) is a one-point set. The idea is that such functors should come from deformations over Artin rings of a fixed object over k. We will call such functors predeformation functors. Example 2.1. In the situation that we have a moduli functor F̃ , we can obtain a predeformation functor F by fixing an element η0 ∈ F̃ (k), and defining F (A) = {η ∈ F̃ (A) : η|k = η0}. In the case that F̃ is representable, the above will give a particularly well-behaved predeformation functor, but in general, to go between a global moduli problem and an associated deformation functor, we will want to do something slightly different, which cannot be expressed as well on the level of functors (however, since plunging into stacks would be a bit much, we settle for an example to give the general idea). Example 2.2. Let X0 be a variety over k. We define a functor F , the functor of deformations of X0, as follows: F (A) is the set of schemes X, flat over Spec A, together with a morphism X0 → X such that the diagram X0 //  X  Spec k // Spec A commutes, and such that we have an induced isomorphism X0 ∼ → X ×A k. Note that in this case the morphism X0 → X is a homeomorphism on underlying topological spaces, so that X differs from X0 only on the level of structure sheaves. Definition 2.3. Let F be a predeformation functor. The tangent space TF is defined to be F (k[ǫ]/ǫ2). We say that F is formally smooth if, for all A′ ։ A in Art(k) with kernel a square-zero ideal, and every element η ∈ F (A), there exists an element η′ ∈ F (A′) with η′|A = η. Note that if F happens to be obtained from a functor F̃ represented by some X as in the first example above, then TF is in fact the tangent space of X at the chosen point, and X is formally smooth at the chosen point if and only if F is formally smooth. As suggested above, we will focus on the study of tangent spaces, with some discussion of obstructions, which measure the failure of a predeformation functor to be smooth. A different question, treated systematically in [4], has to do with with how close a predeformation functor F is to being representable in a suitable sense. However, we will not pursue this direction here. Remark 2.4. In older references such as [4], the phrase “deformation functor” is frequently reserved for the functor of deformations of schemes, with the term “func- tor of Artin rings” used more generally. However, there are many different types of A GLIMPSE OF DEFORMATION THEORY 5 the only known way to calculate the fundamental group involves using the classi- cal topological formula for curves over C, applying this to curves in characteristic 0 more generally, and realizing the given curve in characteristic p as the specialization of a curve in characteristic 0 as above. While the above application involved using knowledge of the generic fiber to prove results about X0, one can also work in the other direction, typically by choosing X0 to be singular. This is the essence of degeneration arguments. A typical situation might be as follows: suppose we wish to prove something about curves of genus g, and it turns out to be enough to prove the statement for a single curve of genus g (this is the case, for instance, with the Brill-Noether theorem). One might take smooth curves of genus 1 and g − 1, and glue them together at a single node to form X0. In this case, the smooth generic fiber provided by the above theorem will have genus g, and one can hope to prove the desired statement for the generic fiber by making sense of it for X0, and understanding the situation for X0 in terms of the two smooth components. Since these components each have genus less than g (assuming g > 1), this potentially sets up an induction with base case g = 1. But to even get started, one has to know that the chosen X0 can be put into a family with smooth generic fiber, and that is where the above theorem comes in. We very briefly describe the proof of Winter’s theorem, as it is typical of a certain class of results. The first step is to show that one can construct compatible systems of families over larger and larger Artin rings, for instance, over k[t]/tn for each n. This is where deformation theory is used. One then makes an argument (usually fairly straightforward, but not completely automatic) that one can construct a deformation over the “formal scheme” (see §II.9 of [2]) corresponding to the limit A of the given Artin rings (in the case above, A = k[[t]]). Next, one applies a theorem of Grothendieck to “effective” the deformation, which means to realize it over the standard scheme Spec A. Here, some real conditions arise, and there are many examples of deformations which arise in nature and can be constructed over formal schemes, but not effectivized. Finally, one applies a theorem of Artin to “algebraize” the deformation, approximating it for instance over a curve of finite type over k. Lifting Galois representations. One intriguing application of deformation the- ory has been Mazur’s theory of deformations of Galois representations, and its highly successful application to the proof of the Shimura-Taniyama-Weil conjec- ture on modularity of elliptic curves, and further progress in recent years. Al- though there is no geometry involved, the theory does fall neatly into the setting of predeformation functors. It had been known for a long time that the following problem is important: Question 3.8. Given a Galois representation ρ : GQ̄/Q → GL2(Fp), what are the possible lifts to representations ρ̂ : GQ̄/Q → GL2(Zp)? The idea is that in order to answer this question, one lifts ρ successively to Z/pnZ for increasing n, and then forms the inverse limit to obtain a lift to Zp. The Artin rings in question are the Z/pnZ, and the possible lifts for each n define the predeformation functor. In [3], Mazur considered this problem in the context of Schlessinger’s theory, showing that the resulting deformation theory is well-behaved. 6 BRIAN OSSERMAN This formed an integral part of Wiles’ technique and subsequent work on Galois representations. 4. Sketch of an argument We discuss one more application of deformation theory to a classical question, with a somewhat more detailed description of how deformation theory can be used to prove the stated result. We consider the following special case of the Brill- Noether problem: Question 4.1. Given g ≥ 0, for which d > 0 is it the case that every curve of genus g has a non-constant map to P1 of degree at most d? For g = 0, the only curve is P1 itself, so the answer is that any d will do. For g = 1, we cannot have a map of degree 1 to P1, as such a map would have to be an isomorphism. But we always have a map of degree 2, so any d ≥ 2 is okay. The general statement is the following: Theorem 4.2. The answer to the above question is: any d for which 2d−2−g ≥ 0. The proof of this theorem is in two parts: an existence statement when 2d− 2− g ≥ 0, and a non-existence statement when 2d − 2 − g < 0. We will now sketch a simple proof based on deformation theory for the non-existence statement. We can immediately check the desired statement in the cases g = 0 or g = 1: for g = 0, the assertion is only that d ≥ 1, which is vacuous, while for g = 1, the assertion is that d ≥ 2, which is necessary because a map of degree 1 between smooth proper curves is necessarily an isomorphism, and if g > 0 then C is not isomorphic to P1. We therefore assume g ≥ 2. The basic idea is to consider the following sequence of moduli spaces: X → Y → Mg, where: • Mg is the moduli space parametrizing curves C of genus g; • X is the moduli space of pairs (C, f) of curves C of genus g together with a map f : C → P1 of degree d; • Y is the moduli space of pairs (C, f) as for X , except that we consider (C, f) ∼ (C, f ′) if f and f ′ are related by an automorphism of P1; The argument then works as follows: by definition, a curve C with a map to P1 of degree d is precisely a point of Mg which has (at least) a point of Y mapping to it. It follows that in order for every curve of genus g to have a map of degree at most d to P1, we have to have the map Y → Mg be dominant for some degree d′ ≤ d; in particular, we have to have dimY ≥ dimMg = 3g − 3, with the last equality coming from the earlier theorem. The reason for introducing the space X is that it turns out that the dimension of X is easiest to compute directly. Indeed, we can do this via deformation theory, and we find that with g ≥ 2, we have dimX = 2d + 2g − 2. Then, the automorphism group of P1 is 3-dimensional (indeed, it consists of all invertible maps of the form z 7→ az+bcz+d , with simultaneous scaling producing the same map; see Exercise I.6.6 or Example II.7.1.1 of [2]), so we have dim Y = dimX − 3 = 2d +2g− 5. We thus have dimY < dimMg if and only if 2d − 2 − g < 0, proving the desired non-existence statement. A GLIMPSE OF DEFORMATION THEORY 7 Warning 4.3. The above doesn’t quite make sense, as written. One can write down a correct argument entirely in terms of schemes by replacing Mg with the base for a “modular family” of curves of genus g, which is to say, an étale cover of Mg. The intuition here is that a modular family is like the universal family over Mg, but instead of each curve appearing once, each is allowed to appear finitely many times. The other spaces should then be interpreted in terms of the curves in the modular family, and one can make the argument described below in a precise manner. 5. Deformations of smooth varieties and Cech cohomology In Example 2.2 above, we describe the problem of deforming a variety X0 over k. In general, it is not easy to describe such deformations, but in the case that X0 is smooth over k, we have all the tools to at least describe the tangent space of the problem – that is, all deformations of X0 over Spec k[ǫ]/ǫ 2. The first two steps of the analysis are the following results: Proposition 5.1. If X0 is a smooth affine variety over k, every first-order defor- mation of X0 is isomorphic to the trivial deformation X0[ǫ]. Indeed, we will see, with the help of a lemma on flatness, that this is equivalent to Exercise II.8.7 of [2], which was on Problem Set 12, Part 1. Proposition 5.2. Let X0 be a smooth variety over k. Then the sheaf of infinites- imal automorphisms of X0 is the tangent sheaf TX0 . Here, recall that the tangent sheaf is defined for X0 smooth by Hom(Ω 1 X/k,OX). If, as above, X0[ǫ] denotes the trivial deformation of X0 over Spec k[ǫ]/ǫ 2, infinites- imal automorphisms of X0 are automorphisms of X0[ǫ] over Spec k[ǫ]/ǫ 2 which restrict to the identity on X0. The sheaf of infinitesimal automorphisms of X0 is the sheaf associating to an open subset U ⊆ X0 the infinitesimal automorphisms of U . Assuming for the moment these two propositions, if we let X0 be any smooth variety over k, we can analyze the first-order deformations of X0 as follows: suppose X1 is a first-order deformation of X0; by Proposition 5.1, if {Ui} is any affine open cover of X0, we have that X1|Ui is isomorphic to the trivial deformation for each i. Moreover, if we choose trivializations ϕi : X1|Ui ∼ → Ui[ǫ] for each i, then for any i < j, we have a gluing map ϕi,j : Ui,j[ǫ] ∼ → Ui,j [ǫ] obtained from ϕi|Ui,j and (ϕj |Ui,j) −1, where Ui,j := Ui∩Uj . We note that because the data of the deformation X1 includes the map X0 →֒ X1, each ϕi,j gives the identity when restricted to Ui,j , so is an infinitesimal automorphism of Ui,j, and hence a section of TX0 (Ui,j), by Proposition 5.2. We also note that because the ϕi,j come from the gluing data for schemes, we have the cocycle condition ϕi,j ◦ ϕj,k = ϕi,k for all i < j < k (see Exercise II.2.12 of [2]). Also, we have ϕi,j = ϕ −1 j,i for any i < j. Conversely, it is clear that given the data of sections ϕi,j ∈ TX0(Ui,j) for all i < j, which satisfy the cocycle condition for all i < j < k, and have ϕi,j = −ϕj,i for all i < j (switching to additive notation, as we consider the ϕi,j as sections of the sheaf TX0 ), we can glue to obtain an X1 which will be a first-order deformation of X0. Note here that flatness over Spec k[ǫ]/ǫ 2 can be checked locally, and since locally we are starting with the trivial deformation, our modules are visibly free over k[ǫ]/ǫ2, hence flat.