Bogomolov-Tian-Todorov Theorem

I came up with this interesting idea of blogging the proof of what is commonly called the Bogomolov-Tian-Todorov theorem. It will probably be only one post as I lightly sketch the proof. Originally I was going to go into the details, but I’ve spent several hours going through it now, and it is mostly just tedious calculations.

What does the theorem say? In its current generality it says that if {X} is a Calabi-Yau variety (actually a little weaker than that even) over an algebraically closed field of characteristic {0}, then the deformation functor is formally smooth. As a corollary we get that deformations of Calabi-Yau’s are unobstructed. This theorem and proof have quite a tangled history.

The three names attached to it correspond to a complex manifold proof. It only works over {\mathbb{C}} and uses all sorts of analytic stuff including complex structures. It is kind of horrifying to an algebraic geometer and has no hope of generalizing to Calabi-Yau varieties over other fields (including char {0} alg closed ones). Then Ran in 1992 came up with a slightly more algebraic method for proving it over {\mathbb{C}} with jet bundles. Kawamata extrapolated a slick purely algebraic way of proving it from that. It over other fields of characteristic {0} and then Fantechi and Manetti cleaned up a few things from Kawamata and removed a few hypotheses.

The Kawamata-Fantechi-Manetti proof is the one I’m going to sketch for you. Lastly, I’d like to point out a really interesting alternative approach by Iacono and Manetti which uses the idea of {L_\infty} structure and dg Lie algebras to give a very algebraic proof. At some point I’d like to understand that way of doing it. Lurie gave an ICM address about how deformation problems are somehow equivalent to the data of {L_\infty} algebras or something to that effect which might give some partial extensions to positive characteristic (I really don’t know as I haven’t looked into this at all).

As I’ve pointed out, this proof had very concrete geometric origins, but successive generalizations have altered it into basically abstract algebraic formalism. If you look at the mentioned papers in order it won’t seem that weird, but this approach is definitely the most streamlined and clean with the best results. Since I’m not motivating it much I’ve decided to omit the details.

The proof comes in basically two steps. The first purely formal theorem is sometimes called the Kawamata {T^1} lifting theorem. We’ll fix an algebraically closed field, {k}, of characteristic {0}. Let {Art_k} be the category of Artinian local algebras over {k} with residue field {k}. We will call a functor {F: Art_k\rightarrow Set} a deformation functor if it admits a hull and an obstruction theory. The terminology deformation functor suggests that whenever our functor is {Def_X} for some reasonable variety {X/k} it will satisfy these two conditions.

For those who have not seen this before a hull is a prorepresentable functor {G} that maps smoothly {G\rightarrow F}. A hull should be thought of as some sort of weak form of a representing object. If {F} is prorepresentable, then the prorepresenting object is a hull, but in general if it is not prorepresentable, then there could be lots of non-isomorphic hulls. The theorem says that if {F} is a deformation functor that satisfies the {T^1}-lifting condition, then {F} is smooth.

So what is the {T^1}-lifting property? It is the condition that the natural map {F(k[x,y]/(x^{n+2}, y^2))\rightarrow F(k[x,y]/(x^{n+1},y^2))\times_{F(k[t]/(t^{n+1}))} F(k[t]/(t^{n+2}))} is surjective for all natural numbers {n}. A more illuminating way of saying this is the following. If {A_n=k[t]/(t^{n+1})} and {B_n=k[x,y]/(x^{n+1}, y^2)}, and {\beta_n: B_n\rightarrow A_n} sends {x\mapsto t} and {y\mapsto 0}, then for some object {X_n\in F(A_n)} (an {n}-th order deformation of {X} maybe), form the set {T^1(X_n/A_n)=\{Y_n\in F(B_n): F(\beta_n)(Y_n)=X_n\}}. These are the first-order infinitesimal deformations of {X_n} (thinking of smooth varieties) i.e. if {F=Def_X} and {X_n} is a lift to {A_n}, then {T^1(X_n/A_n)\simeq H^1(X_n, T_{X_n/A_n})}.

The {T^1}-lifting property in terms of {T^1} is just that for any {X_{n+1}\in F(A_{n+1})} we have {T^1(X_{n+1}/A_{n+1})\rightarrow T^1(X_n/A_n)} is surjective. Or unwinding a little further, if we can lift {X} to {A_n}, then we can lift it to {A_{n+1}}. You might object at this point that the {T^1}-lifting property stated in these terms is exactly the statement that {F} is unobstructed, but this isn’t true because remember that {A_n=k[t]/(t^n)}. This is a very restricted class of lifting that we have to check and Kawamata’s theorem says this suffices to get unobstructedness for deforming over all local Artin rings.

One idea of the proof is to use a well-known fact that a {k}-algebra {R} is smooth if and only if every map {R\rightarrow A_n} lifts to a map {R\rightarrow A_{n+1}}. Another aspect of the proof is to use the fact that we had to assume an obstruction theory exists for {F}. These two ideas play off eachother, and the rest is basically tedious calculations. You could probably almost piece it together yourself from these hints.

Warning, this really, really only works in characteristic {0}. It isn’t just that there is no known proof, but there are very simple counterexamples. Suppose {k} has characteristic {p>0}. You can check that {F(A)=Hom_{k-alg}(k[t]/(t^p), A)} is a deformation functor that satisfies the {T^1}-lifting property (pretty easy exercise to familiarize yourself with the definitions!), but by construction it is representable by {k[t]/(t^p)} and this is NOT smooth. (There is a nice divided power/crystalline work around to this by Schroer, though…).

Alright. So now we know that in order to check that the deformations of a Calabi-Yau are unobstructed we need only check that {Def_X} is actually a deformation functor, and that it satisfies the {T^1}-lifting property. Again, this isn’t so bad. For instance, {Def_X} is certainly a deformation functor (almost by creation of the term “deformation functor”). The two Calabi-Yau properties that get used are the fact that {\omega_X\simeq \mathcal{O}_X} and that {H^0(X, T_X)=0} (again, that second fact isn’t necessarily true in positive characteristic since it requires use of Hodge symmetry). This finishes the proof of the Bogomolov-Tian-Todorov theorem, namely that Calabi-Yau’s are unobstructed in characteristic {0}.


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