Strict applications of deformation theory in which to dip one's toe

Question

I hesitate to ask a question like this, but I really have tried finding answers to this question on my own and seemed to come up short. I readily admit this is due to my ignorance of algebraic geometry and not knowing where to look... Then I figured, that's what this site is for!

Here's the short of it:

What are some examples of strict applications of deformation theory? That is, what are examples of problems that can be stated without mentioning deformation theory or moduli spaces and one of whose solutions uses deformation theory? Please state the problem precisely in your answer, and provide a reference if at all possible :)

Here's the long of it:

I really want to swim in the Kool-Aid fountain of deformation theory and taste of its sweet, sweet purple love, but I'm having trouble. When I wanted to learn about K-theory, I learned about it through the solution to the Hopf invariant one problem, the solution to the vector fields on spheres problem, and through the Adams conjecture. When I wanted to learn some equivariant stuff, it was nice to have the solution to the Kervaire invariant one problem as a guiding force. I have trouble learning things in a bubble; I need at least a slight push.

Now, I know that deformation theory is useful for building moduli spaces, but the trouble is that, aside from the ones that appear in homotopy theory, I haven't fully submerged in this sea of goodness either. The exception would be any example of a strict application that used deformation theory to construct some moduli space and then used this space to prove some tasty fact.

To give you all an idea, here are the only examples I have found (from asking around) that fit my criteria:

1. Shaferavich-Parshin. Let $B$ be a smooth, proper curve over a field and fix an integer $g \ge 2$. Then there are only finitely many non-isotrivial (i.e. general points in base have non-isomorphic fibers) families of curves $X \rightarrow B$ which are smooth and proper and have fibers of genus $g$.
2. Given $g\ge 0$, then every curve of genus $g$ has a non-constant map to $\mathbb{P}^1$ of degree at most $d$ whenever $2d - 2 \ge g$.
3. There are finitely many curves of a given genus over a finite field.
4. The solution to the Taniyama-Shimura conjecture uses deformations of Galois reps.

1, 2, and 3 are stolen from Osserman's really great note: https://www.math.ucdavis.edu/~osserman/classes/256A/notes/deform.pdf

I really like the theme of 'show there are finitely many gadgets by parameterizing these gadgets by a moduli space with some sort of finite type assumption, then showing no point admits nontrivial deformations.' Any examples of this sort would be doubly appreciated. (I guess Kovács and Lieblich have an annals paper where they do something along these lines for the higher-dimensional version of the Shaferavich conjecture, but since they end up counting deformation types of things instead of things, it doesn't quite fit the criteria in my question... but it's still neat!)

Galois representations are definitely a huge thing, and I'd be grateful for any application of their deformation theory that's more elementary than, say... the Taniyama-Shimura conjecture.

So yeah, that's it. Proselytize, laud, wax poetic- make Pat Benatar proud.

Answer 1

One of my favourite examples is the following theorem, due to S. Mori:

Theorem A. Let $X$ be a smooth complex projective variety such that $-K_X$ is ample. Then $X$ contains a rational curve. In fact, through any point $x \in X$ there is a rational curve $D$ such that $$ 0 < -(D \cdot K_X )\leq \dim X+1.$$

In other words, smooth Fano varieties over $\mathbb{C}$ are uniruled.

The proof of this beautiful result uses deformation theory in a very striking way. The idea is the following. One first take any map $f \colon C \to X$, where $C$ is a smooth curve with a marked point $0$ such that $f(0)=x$.

Now by deformation theory of maps one knows that, if one requires that the image of $0 \in C$ is fixed, the morphism $f$ has a deformation space of dimension at least $$h^0(C, f^*T_X)-h^1(C, f^*T_X) - \dim X = -((f_*C) \cdot K_X)-g(C) \cdot \dim X.$$

So, whenever the quantity $-((f_*C) \cdot K_X)-g(C) \cdot \dim X$ is positive, there must be a non-trivial family of deformations of the map $f \colon C \to X$ keeping the image of $0$ fixed. Then, by another result of Mori known as bend and break, one is able to show that at some point the image curve splits in several components and that one of them is necessarily a rational curve passing through $x$.

Instead, when $-((f_*C) \cdot K_X)-g(C) \cdot \dim X$ is not positive we are in trouble. But here comes another brilliant idea of Mori: let's pass to positive characteristic! In fact, in positive characteristic we may compose $f \colon C \to X$ with (some power of) the Frobenius endomorphism $F_p \colon C \to C$. This increases the quantity $-((f_*C) \cdot K_X)$ without changing $g(C)$ and allows us to obtain a deformation space which has again strictly positive dimension. So, using the argument above (deformation theory of maps + bend and break), for any prime integer $p$ we are able to find a rational curve through $x_p \in X_p$, where $X_p$ is the reduction of $X$ modulo $p$ (for the sake of simplicity I'm assuming that $X$ is defined over the integers).

Finally, a straightforward argument using elimination theory shows that if $X_p$ admits a rational curve through $x_p$ for every prime $p$, then $X$ admits a rational curve through $x$, too.

It is worth remarking that no proof of Theorem $A$ avoiding the characteristic $p$ reduction is currently known.

This kind of argument was first used by Mori in order to prove the following theorem, which settles a conjecture due to Hartshorne:

Theorem B. If $X$ is a smooth complex projective variety of dimension $n$ with ample tangent bundle, then $X \cong \mathbb{P}^n.$

See [S. Mori, Projective manifolds with ample tangent bundle, Ann. of Math. 110 (1979)].

More details about Theorem $A$ (as well as its complete proof) can be found in the books [Debarre: Higher-dimensional algebraic geometry] and [Kollar-Mori, Birational geometry of algebraic varieties].

Answer 2

Here are few well-known examples which are not of algebro-geometric nature, where a problem was solved via a reduction to a deformation problem/moduli space problem:

1. Donaldson's work on intersection forms of smooth simply-connected 4-manifolds (definite forms must be diagonalizable); the moduli space in this case is the space of instantons (self-dual connections).

2. Thurston's work on hyperbolization of Haken 3-manifolds. The moduli space in question was the character variety, i.e., moduli space of $SL(2,C)$-representations of the fundamental group. The problem of hyperbolization was reduced by Thurston to a certain fixed-point problem (actually, two slightly different problems depending on existence of fibration over the circle) for a weakly contracting map and solved this way.

3. Margulis' arithmeticity theorem: Every rreducible lattice in a higher rank semisimple Lie group $G$ is arithmetic. The very first step of the proof (actually, due to Selberg) is to look at the character variety, which is defined over $Z$ and observe that isolated points are fixed by a finite index subgroup of the absolute Galois group. This implies that the lattice is conjugate to a subgroup of $G(F)$, where $F$ is a number field. The moduli space in this case is again a character variety.

4. Any of hundreds (if not thousands) of papers on application of gauge theory to low-dimensional ($\le 4$) topology, or even higher-dimensional topology as in Ciprian Manolescu's recent disproof of the triangulation conjecture.