An algebraic proof of Bogomolov-Tian-Todorov theorem

Abstract

We give a completely algebraic proof of the Bogomolov-Tian-Todorov theorem. More precisely, we shall prove that if X is a smooth projective variety with trivial canonical bundle defined over an algebraically closed field of characteristic 0, then the L-infinity algebra governing infinitesimal deformations of X is quasi-isomorphic to an abelian differential graded Lie algebra.Comment: 20 pages, amspro

Deformations and Obstructions of Pairs (X,D)

“…As in [IM10], we recover this result using the power of the Cartan homotopy construction and the degeneration of the Hodge-to-de Rham spectral sequence associated in this case with the complex of logarithmic differentials Ω * X (log D). As corollary, we obtain an alternative (algebraic) proof, that, in the case of a log Calabi-Yau pair (Definition 1.5), the DGLA controlling the infinitesimal deformations of the pair (X, D) is homotopy abelian (Corollary 4.4).…”

“…For K = C, this was first proved in [GM90], see also [Ma04]. For any algebraically closed field K of characteristic 0, this was proved in a completely algebraic way in [IM10], using the degeneration of the Hodge-to-de Rham spectral sequence and the notion of Cartan homotopy.…”

On some formality criteria for DG-Lie algebras

“…A well known result, which is very useful in deformation theory (see e.g. [13,14,15,18]) is that if f : V W is an L ∞ -morphism, W is homotopy abelian and f 1 1 : H * (V ) → H * (W ) is injective, then also V is homotopy abelian. If W is formal, then the injectivity in tangent cohomology is not sufficient to ensure the formality of V .…”

Deformations of Calabi-Yau manifolds in Fano toric varieties

“…This is the famous Bogomolov-Tian-Todorov Theorem [4,5,27,28]. A more algebraic proof of this fact [17,19,24] shows that the functor Def Z of infinitesimal deformations of Z is smooth too. In particular, the dimension of the moduli space at the point corresponding to Z is the dimension of H 1 (Z, T Z ) , where T Z denotes the tangent bundle of Z.…”