Abstract
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.
Highlights
For two smooth components 1, 2 meeting in a smooth divisor D a folkloristic statement says that a necessary condition for = 1 ∪ 2 to have a smoothing is that the two normal bundles are dual to each other; that is, N / 1 ⊗ N / 2 O
We prove the existence of a smoothing for a toroidal crossing space under mild assumptions
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 settles a conjecture by Danilov
Summary
Theorem 1.1 provides a lot more flexibility than existing smoothing results, notably Friedman’s [17] for surfaces, Kawamata– Namikawa’s [34] for d-semistable Calabi–Yaus and Gross–Siebert’s [21] allowing a singular total space but with much stronger requirements on X; see [51, 52, 53, 26, 36]. Is generated by global sections and X is Calabi–Yau, so Theorem 1.1 provides a smoothing of X. Our results enable the construction of versal Calabi–Yau families and conjecturally a logarithmic Frobenius manifold structure in a formal neighbourhood of the extended moduli space; see [3], [7, Theorem 1.3]. Because the smoothing deformations are constructed via the Batalin–Vilkovisky formalism in the Gerstenhaber algebra of (log) polyvector fields (Subsection 13.1), the connection to homological mirror symmetry can be made via [3], [33]
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