Fano versus Calabi - Yau

Details Fano versus Calabi - Yau Fano versus Calabi - Yau

2003-02-10

arxiv.org/abs/math/0302101v1

Mathematics

Algebraic Geometry

The last lecture of Professor Andrey N. Tyurin (24.02.1940 - 27.10 2002) given at the Fano conference (Turin, October 2002). T

Scientific paper

In this article we discuss some numerical parts of the mirror conjecture. For any 3 - dimensional Calabi - Yau manifold author introduces a generalization of the Casson invariant known in 3 - dimensional geometry, which is called Casson - Donaldson invariant. In the framework of the mirror relationship it corresponds to the number of SpLag cycles which are Bohr - Sommerfeld with respect to the given polarization. To compute the Casson - Donaldson invariant the author uses well known in classical algebraic geometry degeneration principle. By it, when the given Calabi - Yau manifold is deformed to a pair of quasi Fano manifolds glued upon some K3 - surface, one can compute the invariant in terms of "flag geometry" of the pairs (quasi Fano, K3 - surface).

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