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arXiv · 1908.11235

Smoothing toroidal crossing spaces

Simon Felten·Matej Filip·Helge Ruddat
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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 two and prove a Hodge-de Rham degeneration theorem for such log spaces which 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 are potential applications.

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Simon Felten, Matej Filip, Helge Ruddat. 2019-08-29. Smoothing toroidal crossing spaces. https://doi.org/10.1017/fmp.2021.8

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