Mirror Symmetry via Logarithmic Degeneration Data II

By Mark Gross, Bernd S Siebert. 2007

126 citations recorded by the source corpus.

This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/777f09d78afdde6bdb7e4b69f6314df0ea70f0e7

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Citations in the corpus

Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry (2013, background): “ the integrable system with the base B. Its total space can be thought of as the analytic space corresponding to the mirror dual family X∨ t,t→0. Remark 10.3.3 As far as we understand the approach of [GroSie1,2] gives an example of WCS discussed above. In their case Bsing as a ZPLsubset of B. Their ”
Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “Unfortunately, there are many varieties that do not admit a well-understood embedding into a toric variety, such as the Deligne–Knudsen–Mumford moduli spaces Mg,n of stable n-marked curves for g > 0 (see [14] and [34]) or the toric degenerations arising in the GrossSiebert approach to mirror symmetr”
Dirichlet Branes and Mirror Symmetry (2009, background): “(Lacking of course is a fiber over 0, and ideally one would like to have a degenerate Calabi-Yau over 0, but that is a story for another day: see [205, 206, 207, 208].”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, background): “ology. It seems that we were too cautious, and the convergence holds without any modiﬁcation. 37 In [40] we suggested a procedure of assigning rational multiplicities to edges of trees (see also [27],[28]). This leads to the following picture. Consider the total space tot(Γ) of the local system Γ. I”
From real affine geometry to complex geometry (2007, background): “ (Fukaya category) side. Hopefully, this approach will ultimately lead to a proof of Homological Mirror Symmetry. Further justiﬁcation of these statements will have to wait for further work; however, [GrSi3] lays the groundwork for computation of periods. Conventions. We work in the category Schk of”
Symplectic theory of completely integrable Hamiltonian systems (2011, unknown): “The best con text to approach these singularities is probably algebraic geometry, and we hope that this article may bring s ome additional interactions between algebraic geometers and specialists on integrable systems; interact ions have began to develop in the context of mirror symmetry, where the ”
Theta functions on varieties with effective anti-canonical class (2016, background): “rmations of certain degenerate K3 surfaces with allirreducible components copies ofP2 [GHKS], but ingeneral it is important to include also non-trivial locally trivial11 deformations. For example, in [GrSi3], §5.2 we describe a locally trivial family X0 parametrized by the algebraic torus Spec k[Γ] ”