Towards Mirror Symmetry for Varieties of General Type
By Mark Gross, L. Katzarkov, Helge Ruddat. 2012
71 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5200f553f31eeb6c400ec093139c86e5ff4509fa
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In Geometry and Topology
- D. Auroux
- L. Katzarkov
- M. Abouzaid
- Nick Sheridan
- P. Seidel
- Alexander I. Efimov
- Dmitri Orlov
- I. Smith
In Mathematical Physics
- D. Auroux
- L. Katzarkov
- M. Abouzaid
- Nick Sheridan
- P. Seidel
- Alexander I. Efimov
- Dmitri Orlov
- I. Smith
Citations in the corpus
- Determinantal Barlow surfaces and phantom categories (2012, background): “or Symmetry (HMS) conjecture, originally proposed for Calabi{Yau, and a little later also for Fano varieties - see [AKO]. The scope of HMS was then extended to varieties of general type in [KKOY] and [GKR]. As noted ibid. we have a behaviour of Fukaya categories similar to derived categories for man”
- Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, background): “ mirrors to various related spaces. These results provide 4 MOHAMMEDABOUZAID,DENISAUROUX,ANDLUDMILKATZARKOV a geometric foundation for mirror constructions that have appeared in the recent literature [16, 33, 32, 47, 49, 3, 25]. We focus primarily on the case where V is affine, and other cases which c”
- Homological mirror symmetry for punctured spheres (2011, background): “recent (and perhaps unexpected) development first proposed by the fourth author is that mirror symmetry also extends to varieties of general type, many of which also admit mirror LandauGinzburg models [22, 4, 20]. The first instance of homological mirror symmetry in this setting was established for th”