Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

Abstract

In this paper, we construct a differential graded Batalin-Vilkovisky (dgBV) algebra P V * , * (X) associated to a possibly degenerate Calabi-Yau variety X equipped with local deformation data. This gives a singular version of the (extended) Kodaira-Spencer dgLa in the Calabi-Yau setting. We work out an abstract algebraic framework and use a local-to-global Čech-de Rhamtype gluing construction. Under the Hodge-to-de Rham degeneracy assumption as well as a local assumption that guarantees freeness of the Hodge bundle and applying standard techniques in BV algebras [35,32,51], we prove an unobstructedness theorem, which can be regarded as a singular version of the famous Bogomolov-Tian-Todorov theorem [3,52, 53], in the spirit of the work of 32]. In particular, we recover the existence of smoothing for both log smooth Calabi-Yau varieties (studied by Friedman [16] and ) and maximally degenerate Calabi-Yau varieties (studied by and ). We also demonstrate how our construction can be applied to produce a logarithmic Frobenius manifold structure on a formal neighborhood of the extended moduli space using the technique of 1].

Log Smooth Deformation Theory via Gerstenhaber Algebras

“…Instead we perform a Thom-Whitney resolution yielding a bigraded Gerstenhaber algebra TW(G • Vα S ) with a non-trivial differential d ∶ TW p,q → TW p,q+1 . Once we forget that differential, there is a unique gluing PV X0 S of the TW(G • Vα S ) up to isomorphism, essentially corresponding to [2,Lemma 3.21]. Our proof shows the cohomological principles behind the explicit-constructive proof in [2, Lemma 3.21].…”

“…We provide the dgla in the log setting by translating ideas of Chan-Leung-Ma in [2] to our setting and bridging the remaining gap to log smooth deformation theory. To achieve this k Q -linear dglas do not suffice.…”

“…Spec (N → k[t] (t 2 )). Instead of dglas we employ k Q -linear predifferential graded Lie algebras, a notion which essentially coincides with almost dgla in [2]. We define them and study systematically their basic properties in the Appendix 9.2.…”

“…In the next step the abstract setup of [2] enters. The Gerstenhaber algebra G • X0 S0 corresponds to 0-th order data in [2, Defn.…”

“…The Gerstenhaber algebra G • X0 S0 corresponds to 0-th order data in [2, Defn. 2.9], and the G • Vα S correspond to the higher order data in [2,Defn. 2.13].…”