Towards mirror symmetry for varieties of general type

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau–Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fib...

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Bibliographic Details
Published in:	Advances in mathematics (New York. 1965) Vol. 308; pp. 208 - 275
Main Authors:	Gross, Mark, Katzarkov, Ludmil, Ruddat, Helge
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 21-02-2017
Subjects:	General type varieties Landau–Ginzburg models Mirror symmetry Mixed Hodge theory General type varieties Landau–Ginzburg models Mixed Hodge theory Mirror symmetry
Online Access:	Get full text
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Summary:	The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau–Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau–Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main result shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf.
ISSN:	0001-8708 1090-2082
DOI:	10.1016/j.aim.2016.03.035