Towards mirror symmetry for varieties of general type

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• 作者：Mark Gross^a ; ¹ ; ^{mgross@math.ucsd.edu} ; Ludmil Katzarkov^b ; ² ; ^{lkatzark@math.uci.edu} ; Helge Ruddat^c ; ³ ; ^{ruddat@uni-mainz.de}
• 关键词：Mirror symmetry ; Landau&ndash ; Ginzburg models ; General type varieties ; Mixed Hodge theory
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：21 February 2017
• 年：2017
• 卷：308
• 期：Complete
• 页码：208-275
• 全文大小：1087 K
• 卷排序：308

文摘

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.