New Developments in Stability and Moduli

July. 18-20, 2011. CMS, Zhejiang University




Monday (July 18):
Chair:C.S. Seshadri
9:30-10:30: V. Balaji
11:00-12:00: Y. Hu
Lunce break
1:30-2:30: V. Mehta

Introductory lecture: (only students attend)
2:45-4:15: Yi Hu
4:30-6:00: Xiaowei Wang

Tuesday (July 19):
Chair: R. Thomas
9:30-10:30: A. Bayer
11:00-12:00: X-W. Wang
Lunch breqak
Chair: K-F. Liu
1:30-2:30: S-T. Yau
2:45-3:45 K. Yoshioka
4:15-5:15: X-T. Sun


Wednesday (July 20):
Chair: J. Li
9:00-10:00: Y-H. Kiem
10:30-11:30: Z-B. Qin


Vikraman Balaji: Parahoric bundles and parabolic bundles

Description: In this talk I will discuss some recent work of mine with Seshadri. Let $X$ be an irreducible smooth projective algebraic curve of genus $g \geq 2$ over the ground field $\bc$ and let $G$ be a semisimple simply connected algebraic group. We introduce the notion of a {\em semistable and stable parahoric} torsor under a certain Bruhat-Tits group scheme $\mathcal G$, construct the moduli space of semistable parahoric $\mathcal G$--torsors and identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of $G$. The results give a complete generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles.


Arend Bayer: Bridgeland stability conditions on threefolds and a conjectural
Bogomolov-Gieseker inequality

Description: I will explain a conjectural construction of Bridgeland stability conditions for the derived category any smooth projective threefold. It is based on a conjectured Bogomolov-Gieseker type inequality for the third Chern character of "tilt-stable" two-term complexes.

The conjectured Bogomolov-Gieseker inequality turns out to be interesting by itself, as it implies a Reider-type theorem on freeness and very ampleness of adjoint line bundles, and a version of Fujita's conjecture. This is based on joint work with Aaron Bertram, Emanuele Macri and Yukinobu Toda.


Yi Hu, Modular desingularization of moduli of genus-two stable maps

Description: For the moduli space of genus-g stable maps into a projective space, we describe the local structures of certain associated canonical derived objects and hence that of the moduli space itself, using some moduli parameters. For genus-two (as well as genus-one), this enables us to provide an explicit modular desingularization of the moduli of stable maps.


Young-Hoon Kiem: Moduli spaces of quasi-maps with perfect obstruction theories

Description: A quasi-map to projective space P^{n-1} refers to a line bundle over a curve together with n ordered sections. I will introduce the notion of delta-stable quasi-maps and show that delta-stable quasi-maps form a proper separated Deligne-Mumford stack for each value of delta>0 except for a finite set of walls. I will also talk about the Guffin-Sharpe-Witten models and the wall crossing behaviors.


Vikram Mehta: Stratified Bundles on Simply Connected Varieties in char p

Description: Gieseker had conjectured that there are no stratified bundles
on simply connected varieties in char p. We prove this conjecture, using
the boundedness theorems of Langer and the work of Hrushovski on the first
order theory of the Frobenius automorphism. This is a joint work with Esnault.


Ian Morrison: GIT and birational geometry of moduli spaces of curves

Description: In the mid-1970s, Gieseker explained how to construct the moduli space $\overline{M}_g$ of stable curves as a GIT quotient of the locus of pluricanonical Hilbert points. After a long hiatus, variations on his ideas have recently reemerged as tools for studying questions about the birational geometry of $\overline{M}_g$ and related moduli spaces of pointed curves and maps. In the talk, I will review the applications to the log minimal model program and the $F$-conjectrue.


Zhenbo Qin: Gieseker-Uhlenbeck morphisms and extremal Gromov-Witten invariants

Description: Using techniques of Kiem-Jun Li regarding localized virtual fundamental cycles in Gromov-Witten theory, we determine 1-point extremal Gromov-Witten invariants of the Gieseker moduli spaces of sheaves over the projective plane.


Xiaotao Sun: Stratified bundles and ètale fundamental group

Description: This is a joint work with H. Esnault. Let X be a smooth projective variety
over an algebraically closed field of characteristic p>0, we show the following:
(1) the commutator of etale fundamental group is a pro-p-group if and only if all of irreducible straitified bundles on X has have rank one.
(2) the category of stratified bundles on X is semi-simple with irreducible objects of rank one if and only if the etale fundamental group of X is abelian without p-power quotient.


Xiaowei Wang: Hilbert Mumford Criterion of nodal curve

Description: We prove the GIT stability of connected nodal curve by directly apply Hilbert Mumford criterion. Also several applications will be discussed.


Shingtung Yau: Non-Kahler Calabi-Yau Manifolds

Description: We will discuss the existence of special metrics on bundles over non-Kahler Calabi-Yau threefolds.


Kota Yoshioka: Bridgeland stability conditions and Fourier-Mukai transforms

Description: Bridgeland stability condition is preserved under the Fourier-Mukai transform by its definition. I will explain the relation with Gieseker stability. In particular, I will explain known results on the birational maps of moduli spaces by using Bridgeland stability condition.


Introductory Lectures

Yi Hu: An introduction to Geometric Invariant Theory

Description: In the first part of the talk, I will provide motivations, basic definitions and some general theorems (including the variation of Geometric Invariant Theory quotients), using elementary examples. In the second part of the talk, I will focus on the applications, mainly on factorization Theorem in birational geometry.


Xiaowei Wang: Numerical criterion of stability

Description: I show how numerical criterion of stability can lead to the proof that SLOPE stable bundle are GIT stable, and SMOOTH curves are (GIT) Hilbert stable.

Meng Chen
Hélène Esnault
Jun Li
Sheng-Li Tan
Eckart Viehweg
Meng Chen
Jun Li
Kefeng Liu
Sheng-Li Tan
Workshop sponsored by: Zhejiang University, East China Normal University, Fudan University
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