Formal Abel–Jacobi Maps

Abstract

We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of which the classical Abel-Jacobi map is a special example.

Algebraic models of local period maps and Yukawa algebras

“…If V is a complex of vector spaces and F ⊂ V is a subcomplex we can define the functor G V,F as The main reason of taking quotient for the automorphisms which are homotopy equivalent to the identity is that in this way the functor G V,F is homotopy invariant; this means that every morphism of pairs α : (V, F ) → (W, H) such that both α : V → W and α : F → H are quasiisomorphisms, induces an isomorphism of functors G V,F ≃ G W,H . This fact is essentially proved in [16] (especially in the ArXiv versions); a more detailed study will appear in the forthcoming paper [17]. A first consequence of the homotopy invariance is that if H * (F ) → H * (V ) is injective, then the natural transformation…”

“…The importance of homotopy fibres in deformation theory has been clarified in several places, see e.g. [15,17,30], since they are the right object for the study of semitrivialized deformation problems.…”

“…The solution to this problem that we adopt here is based on the Fiorenza-Manetti description of the period map given in [16,17]. Very briefly (details in Sections 1 and 4), if A * , * X0 denotes the de Rham complex of X 0 and A 0,p X (Θ X0 ) the space of (0, p)-forms with values in the holomorphic tangent bundle, then a deformation of X 0 over B is induced by a holomorphic family ξ : B → A 0,p X (Θ X0 ) of integrable almost complex structures, defined up to gauge equivalence [18,28].…”

Connections and $L_{\infty}$ liftings of semiregularity maps

“…As suggested in [7], and then clarified in [9,16,28], the correct way to interpretate the semiregularity map is as the obstruction map of a morphism of deformation theories, with target in a product of (formal) intermediate Jacobians: this implies in particular that from the point of view of deformation theory it is more appropriate, for every q, to consider the composition…”

On the abstract Bogomolov-Tian-Todorov Theorem

“…In [IM10], the proof involves L ∞ -algebras and L ∞ -morphisms and it is based on a series of algebraic results that were also applied in other context [IM13,Ia15,BM15,FM16]. In this paper, we review some of these ideas of independent interest to establish the following criterion, that we call Abstract Bomolov-Tian-Todorov Theorem, for homotopy abelianity of a DG-Lie algebra (Theorem 3.3).…”