Quantum Cohomology and Birational Geometry

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Proposal: DMS-0072282Abstract: Quantum cohomology can be thought of as an algebraic invariant structure of geometric objects called symplectic manifolds. A birational transformation can be thought of as a transformation of a symplectic manifold that is an isomorphism outside of a tiny region where discontinuity can happen. The motivation of this project is the following question: What is the canonical transformation of quantum cohomology? Namely, what kind of transformation of symplectic manifolds will preserve quantum cohomology? The principal investigator's early results indicate that birational transformations may play the role of canonical transformation. This project is devoted to the systematic study of this phenomenon.