Smoothing toroidal crossing spaces

Abstract

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.

The degeneration formula for stable log maps

“…A scheme of the form X = X 1 D X 2 permits a lift to a log smooth morphism X → k if and only if T 1 := Ext( X , O X ) has a nowhere vanishing section. More generally, if T 1 has a section with smooth zero locus then X can be upgraded to a log toroidal morphism X → k, see [13].…”

Log Smooth Deformation Theory via Gerstenhaber Algebras

“…) and -to satisfy equation (4) -we have ∆ ℓ−1 (v •;i0...in+1 ) = vi0...in+1 where vi0...in+1 is constructed from (v • ) (up to order n) by the right hand side of equation (4). Given (F •;i0...in+1 ) this is an easy diagram chase.…”

“…it should also apply to the deformation theory of log toroidal families of [4] which are a generalization of log smooth families that need not be log smooth everywhere. The key missing step for the general situation is local uniqueness of deformations which is only known for some types of log toroidal families, see [4,Thm. 6.13].…”

“…Acknowledgement. This paper is a byproduct which arose from studying [2] in order to apply it in [4]. I owe many ideas of the present paper to [2].…”

“…I owe many ideas of the present paper to [2]. I thank my collaborator Matej Filip of [4] for pointing me to the subject and especially the idea to control logarithmic deformations by some sort of dgla. I thank my PhD advisor Helge Ruddat for encouraging this work and for constant support.…”