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 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.

arXiv commentsmall update, final version will appear as open access in Forum of Mathematics Pi

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Cited by in corpus (10)

• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing
• Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces
• A tropical view on Landau-Ginzburg models
• Smoothing pairs over degenerate Calabi-Yau varieties
• Tangent curves to degenerating hypersurfaces
• Smoothing Calabi-Yau toric hypersurfaces using the Gross-Siebert algorithm
• Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program
• Smoothing, scattering, and a conjecture of Fukaya
• Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large $b_2$