Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Mirror symmetry via logarithmic degeneration data, II

Authors: Mark Gross and Bernd Siebert
Journal: J. Algebraic Geom. 19 (2010), 679-780
Published electronically: June 22, 2010
MathSciNet review: 2669728
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Abstract | References | Additional Information

Abstract: 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-Yau spaces. 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 certain 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.

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Additional Information

Mark Gross
Affiliation: UCSD Mathematics, 9500 Gilman Drive, La Jolla, California 92093-0112

Bernd Siebert
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstraße 1, 79104 Freiburg, Germany
Address at time of publication: Department Mathematik, Bundesstraße 55, 20146 Hamburg, Germany

Received by editor(s): September 14, 2007
Received by editor(s) in revised form: December 8, 2009
Published electronically: June 22, 2010
Additional Notes: This work was partially supported by NSF grant 0505325 and DFG priority programs “Globale Methoden in der komplexen Geometrie” and “Globale Differentialgeometrie”.

American Mathematical Society