Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry.(English)Zbl 0864.14022

The authors propose the definition of the “string-theoretic Hodge numbers” for a compact algebraic variety $$X$$ over $$\mathbb{C}$$ with at most Gorenstein quotient or toroidal singularities. Then they prove that if $$X$$ admits a smooth crepant toroidal desingularisation then these Hodge numbers coincide with the usual Hodge numbers of the desingularization. The authors conjecture that “toroidal” is not important for this fact. Mirror duality, Poincaré duality, physicist’s Hodge numbers of orbifolds are discussed and the connection of them with string-theoretic Hodge numbers is conjectured in general and established for specific cases.
The authors have in mind “to convince the reader of the existence of some new cohomology theory $$H^*_{st}(X)$$” which is called the string cohomology of $$X$$, that the string-theoretic Hodge numbers would be the part of.

MSC:

 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14F99 (Co)homology theory in algebraic geometry 14D07 Variation of Hodge structures (algebro-geometric aspects) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 17B55 Homological methods in Lie (super)algebras
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