(Semi)simple exercises in quantum cohomology

Details (Semi)simple exercises in quantum cohomology (Semi)simple exercises in quantum cohomology

2001-03-26

arxiv.org/abs/math/0103164v1

The Fano Conference, Univ. Torino, Turin, 2004, 143-173

Mathematics

Algebraic Geometry

30 pp., amstex file, no figures

Scientific paper

The paper is dedicated to the study of algebraic manifolds whose quantum cohomology or a part of it is a semisimple Frobenius manifold. Theorem 1.8.1 says, roughly speaking, that the sum of $(p,p)$--cohomology spaces is a maximal Frobenius submanifold that has chances to be semisimple. Theorem 1.8.3 provides a version of the Reconstruction theorem, assuming semisimplicity but not $H^2$--generation. Theorem 3.6.1 establishes the semisimplicity for all del Pezzo surfaces, providing an evidence for the conjecture that semisimplicity is related to the existence of a full system of exceptional sheaves of the appropriate length. Finally, in \S 2 we calculate special coordinates for three families of Fano threefolds with minimal cohomology.

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