Algebraic Geometry Seminar

The first part of the talk will be an expository survey of the Gopakumar-Vafa (GV) invariants of a Calabi-Yau threefold. The GV invariants are "virtual counts" of genus g curves in a fixed curve class. They are the best such invariants in that they conjecturally: (1) most accurately reflect the content of genus g curves in the class; (2) are zero for all but finitely many g in a fixed class; and (3) they underlie all other curve-counting theories people may have heard of (Gromov-Witten, Donaldson-Thomas, Pandharipande-Thomas). My main example will be a local K3 surface. In the second part of the talk, I will describe work in progress with Jim Bryan where we develop the theory of GV invariants for certain orbifold Calabi-Yau threefolds. I'll give formulas for the invariants in terms of modular forms and theta functions for the case of local orbifold K3 surfaces.
Speaker(s): Stephen Pietromonaco (University of Michigan)