# Difference between revisions of "Symplectic Geometry Seminar"

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− | We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus has finite dimensional solution space assuming G acts on X with | + | We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus has finite dimensional solution space assuming G acts on X with finitely many orbits. When X is a partial flag variety, we show that this system gives the Picard-Fuchs system of Calabi-Yau hypersurfaces in X. |

In particular, when X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which have played important roles in studying periods of Calabi-Yau hypersurfaces/complete intersections in toric varieties. | In particular, when X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which have played important roles in studying periods of Calabi-Yau hypersurfaces/complete intersections in toric varieties. |

## Revision as of 11:03, 15 September 2011

Wednesday 3:30pm-4:30pm VV B139

- If you would like to talk in the seminar but have difficulty with adding information here, please contact Dongning Wang

date | speaker | title | host(s) |
---|---|---|---|

Sept. 21st | Ruifang Song | The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties | |

Sept. 28th | Dongning Wang | Seidel Representation for Symplectic Orbifolds |

## Abstracts

**Ruifang Song** *The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties *

Abstract

We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus has finite dimensional solution space assuming G acts on X with finitely many orbits. When X is a partial flag variety, we show that this system gives the Picard-Fuchs system of Calabi-Yau hypersurfaces in X.

In particular, when X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which have played important roles in studying periods of Calabi-Yau hypersurfaces/complete intersections in toric varieties.

In general, suppose X is a Fano variety, L is the anticanonical line bundle and G=Aut(X), this construction can be used to study the Picard-Fuchs system of Calabi-Yau hypersurfaces/complete intersections in X.