Algebraic models of local period maps and Yukawa algebras

Abstract

We describe some L∞ model for the local period map of a compact Kähler manifold. Applications include the study of deformations with associated variation of Hodge structure constrained by certain closed strata of the Grassmannian of the de Rham cohomology. As a byproduct we obtain an interpretation in the framework of deformation theory of the Yukawa coupling.

Formal Abel–Jacobi Maps

“…l 0 +i 0 [1] as in Remark 4.14. If the morphism g 000 → g 001 is the inclusion of a DG-Lie subalgebra and g 1•• is a pullback of inclusions of DG-Lie algebras, then the induced morphism at the cohomology level…”

“…Consider M [−1] as a differential associative algebra with trivial product, then the linear map ∞ : T (B(A, M ) [1]) → M of degree 0, defined by setting ∞ : R ⊗n 1 ⊗ S 1 → M, n ≥ 0, ∞ a 1 q 1 (t)dt ⊗ · · · ⊗ a n q n (t)dt ⊗ mp(t)dt = a 1 · · · a n m n+1 q 1 ⊗ · · · ⊗ q n ⊗ p , Proof. Let's denote by the same letter d the differentials of A, R, M and S. We shall denote by Q = Q 1 + Q 2 the differential on the reduced tensor coalgebra Notice that for x ∈ A ⊕ M the degree of xp(t)dt in B(A, M ) [1] is equal to the degree of x in A ⊕ M . We need to prove that d ∞ = ∞ Q.…”

Formality conjecture for minimal surfaces of Kodaira dimension 0

“…the integer α(σ) is even for p = 2 and χ(σ)(−1) α(σ) = 1 if |ξ i | is odd for every i. Formulas (2.3) and (2.4) are well known and essentially date back to Kadeishvili's paper [Kad82]: the choice of signs comes from standard décalage isomorphisms applied to the explicit formulas used in [BM18,Theorem 3.7] and [Man20].…”

“…Denoting by and the inclusions and the projections given by the splitting , we define the maps that satisfy the contraction identities Then a minimal algebra and an extension of to an quasi-isomorphism are defined by the recursive equations where Notice that for every we have the integer is even for and if is odd for every . Formulas (2.3) and (2.4) are well known and essentially date back to Kadeishvili's paper [Kad82]: the choice of signs comes from standard décalage isomorphisms applied to the explicit formulas used in [BM18, Theorem 3.7] and [Man20].…”

$L_{\infty}$ liftings of semiregularity maps via Chern-Simons classes

“…By cofreeness of S(M [1]), the correspondence sending F to its corestriction f = pF , where we denote by p : S(M [1]) → M [1] the natural projection, establishes a bijection between the set of morphisms of graded coalgebras F : S(L [1]) → S(M [1]) and the set of morphism of graded vector spaces f : S(L [1]) → M [1]: in general, compatibility with the bar differentials translates into a countable sequence of algebraic equations in f , see e.g. [3,18]. However, in the particular situation we are concerned with, that is, when the bracket on M is trivial, the situation simplifies considerably, and we have that f ∈ Hom 0 K (S(L [1]), M [1]) is the corestriction of an L ∞ morphism F : S(L [1]) → S(M [1]) if and only if r 1 f = f Q, where we denote by r 1 the shifted differential r 1 (m) = −d M (m) on M [1] and by Q the bar differential on S(L [1]).…”