Algebra and Geometry Seminar

Quantum cohomology is a deformation of the cohomology ring defined by counting rational curves. A close relationship between quantum cohomology and birational geometry has been expected. For example, when the quantum parameter q approaches an "extremal ray", the spectrum of the quantum cohomology ring clusters in a certain way (predicted by the corresponding extremal contraction), inducing a decomposition of the quantum cohomology. In this talk, I will discuss such a decomposition for blowups: quantum cohomology of the blowup of X along a smooth center Z will decompose into QH(X) and (codim Z-1) copies of QH(Z). The proof relies on Fourier analysis and shift operators for equivariant quantum cohomology.  We can describe blowups as a VGIT of a certain space W with C^* action. Then the equivariant quantum cohomology of W acts as a "global" mirror family connecting X and its blowup.