UCSD Algebraic Geometry Seminar, Spring 2018



Meetings are held on Fridays 2:30-3:30 in AP&M 5829. A pre-talk is held from 2:00-2:30.

Schedule
April 13 Roberto Svaldi (University of Cambridge)
Title: Birational boundedness of rationally connected klt Calabi-Yau 3-folds
Abstract: Calabi-Yau varieties and Fano varieties are building blocks of varieties in the sense of birational geometry. Birkar recently proved that Fano varieties with bounded singularities belong to just finitely many algebraic families. One can then ask if an analogous result holds for Calabi-Yau varieties. If one only considers rationally connected Calabi-Yau varieties with klt singularities - those Calabi-Yau varieties behaving most like Fano - Shokurov actually conjectured that also these varieties should be bounded in any fixed dimension. We show that rationally connected klt Calabi-Yau 3-folds form a birationally bounded family. In many cases, we can actually give more precise statements and we are able to relate the boundedness problem to the study of a quite mysterious birational invariant: the minimal log discrepancy. This is a joint work in progress with W. Chen, G. Di Cerbo, J. Han, and C. Jiang.
April 14 (Saturday)
The Southern California Algebraic Geometry Seminar takes place at USC.
April 20 Chiara Damiolini (Rutgers University)
Title: Conformal blocks associated with twisted groups
Abstract: Let G be a simple and simply connected algebraic group over a field. We can attach to a it the sheaf of conformal blocks: a vector bundle on M_g whose fibres are identified with global sections of a certain line bundle on the stack of G-torsors. We generalize the construction of conformal blocks to the case in which G is replaced by a "twisted group" defined over curves in terms of covering data. In this case the associated conformal blocks define a sheaf on a Hurwitz stack and have properties analogous to the classical case.
May 18 Martijn Kool (Utrecht University)
Title: New directions in Vafa-Witten theory
Abstract: In the 1990's, Vafa-Witten tested S-duality of N=4 SUSY Yang-Mills theory on a complex algebraic surface X by studying modularity of a certain partition function. In 2017, Tanaka-Thomas defined Vafa-Witten invariants by constructing a symmetric perfect obstruction theory on the moduli space of Higgs pairs (E,φ) on X. The instanton contribution (φ=0) to these invariants is the virtual Euler number of moduli space of sheaves. I outline a method to calculate this contribution, when X is of general type, by reducing to descendent Donaldson invariants. For rank 2, this leads to verifications of a formula from Vafa-Witten. The method can be "refined" to virtual χy genus, elliptic genus, and cobordism class, which involves weak Jacobi forms and Borcherds lifts thereof. I also give a new formula for rank 3 VW invariants on general type surfaces, correcting an error in the physics literature. Joint with Göttsche.
May 25 Alexander Perry (Columbia University)
Title: TBA
Abstract: TBA

For titles and abstracts, please see also the mathematics department calendar.

Organizers: Elham Izadi, James McKernan and Dragos Oprea

This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Past quarters: Fall 2013 , Winter 2014 , Spring 2014 , Fall 2014. Fall 2017. Winter 2018. Contact Jonathan Conder at jconder@ucsd.edu about problems with the website or posters.