Fano versus Calabi - Yau
Abstract
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).
Cite
@article{arxiv.math/0302101,
title = {Fano versus Calabi - Yau},
author = {Andrey N. Tyurin},
journal= {arXiv preprint arXiv:math/0302101},
year = {2007}
}
Comments
The last lecture of Professor Andrey N. Tyurin (24.02.1940 - 27.10 2002) given at the Fano conference (Turin, October 2002). The text of the lecture was prepared and edited by Nik. Tyurin and Yulia Tiourina (PennStateUniv) due to support of Max - Planck - Institute fur Matematik (Bonn, Germany). Will be published in the proceedings of the Fano conference