Mathematics – Algebraic Geometry
Scientific paper
2012-02-18
Mathematics
Algebraic Geometry
79 pages, Several references added in new version
Scientific paper
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 results 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. This exchange can be anticipated from an analysis of Hochschild homology of the relevant categories arising in homological mirror symmetry in this case; we also conjecture that a similar, but different, exchange of dimensions arises from Hochschild cohomology, relating the cohomology of sheaves of polyvector fields on the hypersurface to the cohomology of the critical locus.
Gross Mark
Katzarkov Ludmil
Ruddat Helge
No associations
LandOfFree
Towards Mirror Symmetry for Varieties of General Type does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Towards Mirror Symmetry for Varieties of General Type, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Towards Mirror Symmetry for Varieties of General Type will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-32953