A homology theory for tropical cycles on integral affine manifolds and a perfect pairing

Article Mathematics

Smoothing Calabi-Yau toric hypersurfaces using the Gross-Siebert algorithm

Thomas Prince

Summary: This study explains the formation of a new dataset of Calabi-Yau threefolds using the Gross-Siebert algorithm, and explores their connection with Calabi-Yau toric hypersurfaces with Gorenstein singularities. It also demonstrates how to obtain polarised tropical manifolds by 'smoothing the boundary' of four-dimensional reflexive polytopes.

COMPOSITIO MATHEMATICA (2021)

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Article Mathematics, Applied

Smoothing toroidal crossing spaces

Simon Felten et al.

Summary: In this paper, the existence of a smoothing for a toroidal crossing space under mild assumptions is proven. By linking log structures with infinitesimal deformations, the result is presented in a very compact form for normal crossing spaces. The main approach involves studying log structures that are incoherent on a subspace of codimension 2 and proving a Hodge-de Rham degeneration theorem, settling a conjecture by Danilov. This work demonstrates that the homotopy equivalence between Maurer-Cartan solutions and deformations, combined with Batalin-Vilkovisky theory, can be utilized to obtain smoothings, with potential applications in constructing new Calabi-Yau and Fano manifolds, as well as Frobenius manifold structures on moduli spaces.

FORUM OF MATHEMATICS PI (2021)

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Article Mathematics