Smoothing toroidal crossing spaces
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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 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.
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Forum of mathematics : Pi, 9, Cambridge Univ. Press, Cambridge, 2021, https://doi.org/10.1017/fmp.2021.8