Algebraic Geometry Seminar
Fall 2018 — Tuesdays 3:30  4:30 PM, location LCB 222
Date  Speaker  Title — click for abstract (if available) 
August 21 


August 28 
Patrick Graf University of Bayreuth 
The Kodaira Problem: Results and
Perspectives
The Kodaira problem asks whether every compact Kähler manifold can
be deformed to a projective one. While Voisin gave counterexamples in 2004, a
modified version for nonuniruled spaces remains open, and in fact has been
established in dimension at most three by Claudon, Höring, Lin and myself. I
will review these results and then talk about the (im)possibility of extending
the conjecture to uniruled spaces. If time permits, I will also outline a
current project dealing with higher dimensions. The latter two works are joint
with Martin Schwald (Essen).

September 4 
Harold Blum University of Utah 
Moduli of uniformly Kstable Fano
varieties
In order to have a well behaved moduli functor for Fano varieties, it seems
natural to restrict oneself to Fano varieties that are Kpolystable. Recall,
Kstability is an algebraic notion that characterizes when a smooth Fano
variety admits a KahlerEinstein metric.
In this talk, we consider the behavior of uniform Kstability (a strengthening
of Kstability) in families. We will explain that uniform Kstability is an
open condition in QFano families and the moduli functor of uniformly Kstable
QFano varieties is separated. Together with a boundedness result of C. Jiang,
these results yield a separated DM stack paratmerizing uniformly Kstable Fano
varieties of fixed dimension and volume. This is joint work with Yuchen Liu and
Chenyang Xu.

September 11 


September 18 
Pablo Solis
Stanford University 
Natural Cohomology on P1 x P1
I’ll begin with a discussion of the classification of vector bundles on P1 and
explain what natural cohomology means in this context. Then I’ll consider the
case of vector bundles on P1 x P1. In general vector bundles on surfaces are
more complicated but a useful tool allows one to reduce many problems about
vector bundles to questions of linear algebra. This is the theory of monads.
I’ll discuss monads and show how they are used to prove a conjecture of
Eisenbud and Scheryer about vector bundles on P1 x P1 with natural cohomology.

September 25

Raphael Rouquier
University of California, Los Angeles 
Representation theory on spaces
I will discuss an emerging theory where geometrical objects arising in representation
theory are themselves viewed as "representations of a higher group". More general
"higher groups" should arise as invariants of algebraic varieties and I will speculate
on connections with higher algebraic Ktheory and spaces of stability conditions.
I will describe some aspects of the representation theory of the "higher group"
associated to a point.

October 2 
Ziquan
Zhuang Princeton University 
Birational superrigidity and
Kstability
Birational superrigidity and Kstability are properties of Fano varieties that
have many interesting geometric implications. For instance, birational
superrigidity implies nonrationality and Kstability is related to the
existence of KählerEinstein metrics. Nonetheless, both properties are hard to
verify in general. In this talk, I will first explain how to relate birational
superrigidity to Kstability using alpha invariants; I will then outline a
method of proving birational superrigidity that works quite well with most
families of index one Fano complete intersections and thereby also verify their
Kstability. This is partly based on joint work with Charlie Stibitz and Yuchen
Liu.

October 9 
Fall Break 

October 16 
Stefano Filipazzi University of Utah 
A generalized canonical bundle
formula and applications
Birkar and Zhang recently introduced the notion of generalized
pair. These pairs are closely related to the canonical bundle formula and
have been a fruitful tool for recent developments in birational geometry.
In this talk, I will introduce a version of the canonical bundle formula
for generalized pairs. This machinery allows us to develop a theory of
adjunction and inversion thereof for generalized pairs. I will conclude by
discussing some applications to a conjecture of Prokhorov and Shokurov.

October 23 
Joaquín Moraga University of Utah 
Minimal log discrepancies and
Kóllar components
The minimal log discrepancy of an algebraic variety is an invariant which measures the singularites of the variety.
For mild singularities the minimal log discrepancy is a nonnegative real value; the closer to zero this value is, the more singular the variety.
It is conjectured that in a fixed dimension, this invariant satisfies the ascending chain condition.
In this talk we will show how boundedness of Fano varieties imply some local statements about the minimal log discrepancies of klt singularities.
In particular, we will prove that the minimal log discrepancies of klt singularities which admit an eplt blowup can take only finitely many possible values in a fixed dimension.
This result gives a natural geometric stratification of the possible mld's on a fixed dimension by finite sets.
As an application, we will prove the ascending chain condition for minimal log discrepancies of exceptional singularities in arbitrary dimension.

October 30 


November 6 


November 13 
Lei Wu University of Utah 
Hyperbolicity of base spacs of log
smooth families
People expect moduli spaces to be hyperbolic at least in the sense
of stack. I will discuss known results about both analytic and
algebraic hyperbolicity of certain moduli of varieties, for instance
M_g and families of general type varieties. Then I will talk about
hyperbolicity for base spaces of log smooth families of log general
type pairs. I will use it to prove hyperbolicity of the moduli stack
of Riemann surfaces of genus g with n marked points. This is joint
work with Chuanhao Wei.

November 20 
Renzo Cavalieri Colorado State University 
Witten conjecture for Mumford's
kappa classes
Kappa classes were introduced by Mumford, as a tool to explore the
intersection theory of the moduli space of curves. Iterated use of
the projection formula shows there is a close connection between the
intersection theory of kappa classes on the moduli space of
unpointed curves, and the intersection theory of psi classes on all
moduli spaces. In terms of generating functions, we show that the
potential for kappa classes is related to the GromovWitten
potential of a point via a change of variables essentially given by
complete symmetric polynomials, rediscovering a theorem of Manin and
Zokgraf from '99. Surprisingly, the starting point of our story is a
combinatorial formula that relates intersections of kappa classes
and psi classes via a graph theoretic algorithm (the relevant graphs
being dual graphs to stable curves). Further, this story is part of
a large wallcrossing picture for the intersection theory of Hassett
spaces, a family of birational models of the moduli space of curves.
This is joint work with Vance Blankers (arXiv:1810.11443) .

November 27 
Ziwen Zhu University of Utah 

December 4 
Final Exams 

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