Algebraic Geometry Seminar
Fall 2018 — Tuesdays 3:30  4:30 PM, location LCB 222
Date  Speaker  Title — click for abstract (if available) 
January 9 


January 16 


January 23 


January 30 
Joe Waldron Princeton University 
Singularities of general fibers in positive characteristic
Generic smoothness fails to hold for some fibrations in positive characteristic. We study consequences of this failure, in particularly by obtaining a canonical bundle formula relating a fiber with the normalization of its maximal reduced subscheme. This has geometric consequences, including that generic smoothness holds on terminal Mori fiber spaces of relative dimension two in characteristic p at least 11. This is joint work with Zsolt Patakfalvi.

February 6 
Grigory Mikhalkin
University of Geneva 
Examples of tropicaltoLagrangian correspondence
According to SYZ philosophy the same tropical object
should admit two ways to be lifted classically: as a
complex object, and as a (Tdual) symplectic object.
While the tropicaltocomplex correspondence is
relatively wellstudied, tropicaltosymplectic
correspondence remains significantly less wellstudied
up to date.
In the talk we'll look at some first instances of tropical
tosymplectic correspondence. As an application of such
correspondence in the case of planar tropical curves we'll
reprove a theorem of Givental (from about 30 years ago)
on Lagrangian embeddings of connected sums of Klein
bottles to C^2.
For tropical curves in toric 3folds the resulting Lagrangians
turn out to be Waldhausen graphmanifolds. For this case
we'll relate the enumerative multiplicity of tropical rational curves
to the torsion in the first homology group of the corresponding
Lagrangian submanifolds (in full compliance with Mirror Symmetry
predictions).

February 13

Lei Wu
University of Utah 
Hyperbolicity of base spaces of certain families
Hyperbolicity is an interesting property of varieties both analytically
and algebraically. I will make a recall of several hyperbolicity conjectures
and related results, especially hyperbolicity of moduli spaces. Then
I will discuss hyperbolicity of base spaces of families of smooth general
type varieties, which would imply brody hyperbolicity of some moduli. The
main technical tools we used are Hodge modules and Higgs sheaves. If time
permitted I will discuss how they could be applied to solve the
hyperbolicity problems for families. This is joint work with Mihnea Popa
and Behrouz Taji.

February 20 
Tommaso de Fernex University of Utah 
The complex Plateau problem
The complex Plateau problem has been studied in the context of CR manifolds in works of Rossi (1965) and Harvey and Lawson (1975) where it is proved that every embedded strongly pseudoconvex compact CR manifold of dimension at least 3 is the boundary of a Stein space with isolated singularities. What remains to be determined are intrisic conditions on the CR manifold that guarantee that the Stein filling is smooth. In the talk I will survey this question and discuss some old and recent developments on the problem.

February 27 
Andreas Malmendier Utah State University 
(1,2) polarized Kummer surfaces, elliptic fibrations, and Prym varieties
A smooth K3 surface obtained as the blowup of the quotient of a fourtorus by the involution automorphism at all 16 fixed points is called a Kummer surface. Kummer surface need not be algebraic, just as the original torus need not be. However, algebraic Kummer surfaces obtained from abelian varieties provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail.
In this talk, we give an explicit description for the relation between algebraic Kummer surfaces of Jacobians of genustwo curves with principal polarization and those associated to (1, 2)polarized abelian surfaces from three different angles: the point of view of 1) the binational geometry of quartic surfaces in P^3 using eveneights, 2) elliptic fibrations on K3 surfaces of Picardrank 17 over P^1 using Nikulin involutions, 3) thetafunctions of genustwo using twoisogeny. Finally, we will explain how these (1,2)polarized Kummer surfaces naturally allow for an identification of the complex gauge coupling in SeibergWitten gauge theory with the axiondilaton modulus in string theory using an old idea of Sen. (This is joint work with Adrian Clingher.)

March 9, 1:30 PM JFB B1 (Note time and location changes) 
Yuri Tschinkel Courant (CIMS NYU) 
Joint AG and CA seminar: Rationality and Unirationality
I will discuss
cohomological obstructions to rationality, descent varieties, and
unirationality questions.
Note the speaker is also giving this week's colloquium on March 8. AG seminar participants are encouraged to attend. 
March 13 

Cancelled

March 20 
Spring Break 

March 27 
Yefeng Shen University of Oregon 
LG/CY correspondence for Fermat elliptic curves
We use holomorphic Cayley transformations of quasimodular forms to realize a correspondence between FanJarvisRuanWitten invariants of Fermat elliptic polynomials and GromovWitten invariants of elliptic curves of all genus. Our key observation is that BelorousskiPandharipande’s relation on \bar{M}_{2,3} implies the genus one invariants on both GW theory and FJRW theory are governed by the Chazy equation. The modularity follows from the Chazy equation and the initial values in FJRW theory can be calculated by cosection localization. This work is joint with J Li and J Zhou.

April 3 
Sarah Frei University of Oregon 
Moduli spaces of sheaves on a K3 surface and Galois representations
For a fixed K3 surface, we can study various moduli spaces of sheaves having a given topological type. Under appropriate conditions on the topological data, the moduli spaces are smooth projective varieties. I will discuss a new result about the geometry and arithmetic of these moduli spaces when the K3 surface is defined over an arbitrary field. For any two such moduli spaces of the same dimension, their étale cohomology groups with $\mathbb{Q}_\ell$coefficients are isomorphic as Galois representations. In particular, when the K3 surface is defined over a finite field, this implies that the moduli spaces have equal zeta functions.

April 10 
Roberto Svaldi University of Cambridge 
On the boundedness of CalabiYau varieties in low dimension
I will discuss new results towards the birational boundedness of
lowdimensional elliptic CalabiYau varieties, joint work with Gabriele
Di Certo.
Recent work in the minimal model program suggests that pairs with trivial log canonical
class should satisfy some boundedness properties.
I will show that 4dimensional CalabiYau pairs which are not birational to a product are
indeed log birationally bounded. This implies birational boundedness of elliptically fibered
CalabiYau manifolds with a section, in dimension up to 5.
If time allows, I will also try to discuss a first approach towards boundedness of rationally
connected CY varieties in low dimension (joint with G. Di Cerbo, W. Chen, J. Han and, C. Jiang).

April 17 
Yuchen Liu Yale University 
Kstability of cubic threefolds
We prove the Kmoduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K(semi,poly)stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit KählerEinstein metric as well as provides a precise list of singular KE ones. To achieve this, the main new ingredient is an estimate in dimension three of the normalized volumes of kawamata log terminal singularities introduced by Chi Li. This is a joint work with Chenyang Xu.

April 24, 2:30 PM 
Mirko Mauri University College London 
Dual complexes of log CalabiYau pairs and Mori fibre spaces
Dual complexes are CWcomplexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. They have been finding useful applications for instance in the study of degenerations of projective varieties, mirror symmetry and nonabelian Hodge theory. In particular, Kollár and Xu have asked whether the dual complex of a log CalabiYau pair is always a sphere or a finite quotient of a sphere. It is natural to check first if this holds on the end products of minimal model programs. In this talk, we will give a positive answer for Mori fibre spaces of Picard rank two.

May 1 
Final Exams 

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