April 13 Roberto Svaldi (University of Cambridge) |
Birational boundedness of rationally connected klt Calabi-Yau 3-folds |
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) |
Conformal blocks associated with twisted groups |
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) |
New directions in Vafa-Witten theory |
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 χ_{y} 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) |
Deformation and derived equivalent but non-birational Calabi-Yau threefolds |
I will construct a pair of Calabi-Yau threefolds which are deformation and derived equivalent, but not birationally equivalent. I will explain how this gives a counterexample to the birational Torelli problem for Calabi-Yau threefolds, as well as new examples of zero divisors in the Grothendieck ring of varieties. This is joint work with Lev Borisov and Andrei Caldararu. |
June 1 Michael McQuillan (University of Rome Tor Vergata) |
Very functorial, very easy and very quick resolution of singularities |
June 8 Morgan Brown (University of Miami) |
The Skeleton of a Product of Degenerations |
The essential skeleton is an invariant of a degeneration that appears in both Berkovich geometry and minimal model theory. I will show that for degenerations with a semistable model, the essential skeleton of a product of degenerations is the product of their skeleta. As an application, we are able to describe the homeomorphism type of some degenerations of hyperkähler varieties, both for a Hilbert scheme of degenerating K3 surfaces and for a Kummer variety associated to a degeneration of an abelian surface. This is joint work with Enrica Mazzon. |
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.