Back to Algebraic Geometry Seminar: Spring 2025
Algebraic Geometry Seminar
Date: Tuesday, Feb 25, 2025
Time: 3:30PM - 4:30PM
Location: LCB 225
Mark Shoemaker
Colorado State University
Title |
The Quantum Spectrum and Gamma Structure for Standard Flips |
Abstract |
In his 1998 ICM talk, Dubrovin conjectured that for a Fano manifold X, the derived category of X possesses a full exceptional collection if and only if the quantum cohomology of X is generically semi-simple, suggesting a deep connection between the derived category of X and its Gromov–Witten theory. This relationship has been clarified in recent years, with a prominent role being played by the quantum spectrum, that is, the eigenvalues of the operator c_1(X) q - on H^(X) where c_1(X) is the first Chern class of TX and *q denotes the quantum product at q. Kontsevich has conjectured that the quantum spectrum of X is closely related to semi-orthogonal decompositions of D^b(X). I will describe how, in the case of projective bundles, blow-ups, and standard flips, known semi-orthogonal decompositions of the derived category indeed correspond to the quantum spectrum at a special value of q. This is based on joint work with Yefeng Shen. |
Algebraic Geometry