Mirror Symmetry

Xinyun Zhu (UIUC), The Hilbert scheme of coherent sheaves

Abstract: We will study Chapter 4 in Lectures on Vector Bundles by Le Potier. We will construct the Hilbert scheme of coherent sheaves on a curve which are quotients of a fixed locally free sheaf and have a fixed Hilbert polynomial. This scheme is characterized by a universal property which makes indispensable the introduction of flat families of coherent sheaves parametrized by an algebraic variety S. When S is reduced, this notion can be characterized in terms of global invariants and simply means that the Hilbert polynomial is constant on fibers. When S is not reduced, we can characterize the flatness by introducing direct image sheaves.

Jonathan Cox (UIUC), Enumeration of rational curves via torus actions: More details

Abstract: Today we will continue the localization computations from last Friday. We have seen how the contribution of the normal bundle of a fixed component breaks up into several pieces. With help from the intersection theory of stable curves, we will compute each piece. Then, for each of the three examples described last time, we will compute for each fixed component the contribution of the relevant polynomial in Chern classes to the integral. The final result is a sum over fixed components that can be added up as described last time.

Jonathan Cox (UIUC), Enumeration of rational curves via torus actions: Examples

Abstract: Filling in a gap from last week's talks, we will explain in detail various examples of the enumerative geometry of rational curves. In particular, we will show how the enumerative numbers in each example correspond to integrals of polynomials in Chern classes. The three examples are degree d rational plane curves through 3d-1 general points, degree d rational curves on a quintic threefold, and the contribution of multiple covers of a rational curve to a Gromov-Witten invariant of a Calabi-Yau threefold. If time permits, we will also discuss how to add up the sums resulting from localization of the above integrals. This is done by identifying each sum with the critical value of a certain functional.

Jonathan Cox (UIUC), Enumeration of rational curves via torus actions: The details

Abstract: We will continue to study this groundbreaking paper of Kontsevich. Today's talk will be concerned with the details of the localization computations, especially as they apply to the three enumerative examples in the paper. We will describe the normal bundles of each fixed component under the torus action, as well as the restriction of the appropriate bundles to these components. Once we have reduced the integrals to sums over fixed components using localization, we will examine Kontsevich's (not quite successful) attempts to evaluate these sums using critical values.

Jonathan Cox (UIUC), Enumeration of rational curves via torus actions

Abstract: Gromov-Witten invariants, moduli spaces of stable maps, and using localization on the latter to compute the former: in some sense they all got their (mathematical) start in this paper. Many of the papers we have covered this semester and previously have been "descendents" of this landmark masterpiece. Today we will focus on Kontsevich's motivation for introducing these concepts, look at three examples of enumerative problems where these methods bear fruit, and possibly start to describe the setup for localization computations. Join us as we look back to where it all began!

Yong Fu (UIUC), Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

Abstract: We will continue to study the paper by Li and Tian for a different construction of virtual fundamental cycles in Gromov-Witten theory using tangent-obstruction complex.

Yong Fu (UIUC), Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

Abstract: We will continue to study the paper by Li and Tian for a different construction of virtual fundamental cycles in Gromov-Witten theory using tangent-obstruction complex.

Yong Fu (UIUC), Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

Abstract: We will study the paper by Li and Tian for a different construction of virtual fundamental cycles in Gromov-Witten theory using tangent-obstruction complex.

NO TALK TODAY

Abstract: Participants may want to attend the Coble Memorial Lecture by David Eisenbud at this time.

Jonathan Cox (UIUC), Gromov-Witten theory, Hurwitz numbers, and Matrix models: Some details

Abstract: We will undertake asymptotic analysis of weighted numbers of branching graphs in order to see where the identities used in the new proof of Kontsevich's Theorem come from. This will involve a a good amount of probability, as it requires a study of random trees. This is a continuation of Wednesday's talk.

Jonathan Cox (UIUC), Gromov-Witten theory, Hurwitz numbers, and Matrix models

Abstract: Last week we saw how Hurwitz numbers can be expressed in terms of certain tautological integrals on moduli spaces of stable curves. Today we will look at the other side of the story in this paper of Okounkov and Pandharipande. After giving another definition of Hurwitz numbers as a weighted count of branching graphs, we will find that considering asymptotics of Hurwitz numbers under both definitions gives a new proof of Kontsevich's Theorem on the tautological integrals. Details of the proof will be put off until Friday. Some discussion of matrix models and integrable hierarchies will also be included to give an idea of the approach Kontsevich used in his original proof.

Jonathan Cox (UIUC), Gromov-Witten theory and Hurwitz numbers: The details

Abstract: We will continue to develop the concepts of Gromov-Witten theory, Hurwitz numbers, and their connections as expounded by Okounkov and Pandharipande. In particular, we will explore the Gromov-Witten theory of P^1, briefly review virtual localization, and describe the path between Hurwitz numbers and Hodge integrals. This is a continuation of Wednesday's talk.

Xinyun Zhu (UIUC), Moduli and semi-stability of vector bundles and coherent sheaves

Abstract: More details from Chapter 5 of Lectures on Vector Bundles will be presented, mixed with more general ideas from the paper Moduli of representations of the fundamental group of a smooth projective variety I by Simpson, which was briefly introduced last Friday. The latter generalizes the concepts of stability, semi-stability, and moduli of vector bundles to the context of coherent sheaves.

Xinyun Zhu (UIUC), Semi-stability of vector bundles

Abstract: We will study Chapter 5 of the book Lectures on Vector Bundles by J. Le Potier. The set of isomorphism classes of algebraic vector bundles of fixed rank and degree cannot be parametrized by an algebraic variety. To get around this problem, we introduce to notion of a semi-stable bundle. The study of bundles which are not semi-stable can be reduced to studying semi-stable bundles using Harder-Narasimhan filtrations, whose successive quotients are semi-stable. The most significant result in the chapter shows that the family of semi-stable bundles with given rank and Chern class is bounded.

Xinyun Zhu (UIUC), Moduli of coherent sheaves and semistability

Abstract: We will discuss the paper "Moduli of representations of the fundamental group of a smooth projective variety I" by Simpson, which has made a very important contribution to the subject of moduli of coherent sheaves and vector bundles with additional structures. The new ideas and techniques introduced in this paper have proved to be very useful in the subsequent work of many authors. The paper has a detailed introduction, explaining the development of the various novel ideas which it contains. A notion of stability for pure-dimensional coherent sheaves on a projective variety is introduced. The concept of pure dimensionality is more general than that of torsion freeness which it replaces. Then Simpson applies geometric invariant theory to construct the moduli of semistable sheaves.

Jonathan Cox (UIUC), Two presentations for the Chow ring of a moduli space of stable maps: The details

Abstract: I will give a more detailed account of the methods used to compute the presentations. This is a continuation of Wednesday's talk.

Jonathan Cox (UIUC), Two presentations for the Chow ring of a moduli space of stable maps

Abstract: A presentation for the Chow ring of a space \bar{M}_g,n(X,\beta) gives a full understanding of its intersection theory, and could also give insight into Gromov-Witten invariant computations. I will give two presentations for the Chow ring of \bar{M}_0,2(P^1,2), the first examples of such presentations for n>1 and d>1. The first presentation is computed via localization and linear algebra. Recently, a second presentation has been formulated in a more geometric fashion. Basics and intuition of computing intersections on the moduli space of stable maps will be explained. This is a talk on my thesis research.

Xinyun Zhu (UIUC), Hodge integrals and degenerate contributions: The details

Abstract: We will continue to study this paper by Pandharipande.

Xinyun Zhu (UIUC), Hodge integrals and degenerate contributions

Abstract: We will discuss this paper by Pandharipande. Hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the Gromov-Witten theory of 3-folds. In the Calabi-Yau case, the contributions are compared to related M-theoretic calculations. In the Fano case, the comtributions suggest new integrality conditions.

Jonathan Cox (UIUC), Localization of Virtual Classes: The details

Abstract: Many subtleties are involved in extending the Atiyah-Bott localization formula to singular Deligne-Mumford stacks admitting an embedding into nonsingular Deligne-Mumford stacks. We will investigate how Graber and Pandharipande tackle some of these issues in their proof of this more general localization formula. We will also explore more details of their Gromov-Witten computations. This is a follow-up to Wednesday's introductory talk.

Jonathan Cox (UIUC), Localization of Virtual Classes

Abstract: We will discuss this paper, in which Graber and Pandharipande extend the Atiyah-Bott localization formula to singular schemes and stacks which admit an embedding into a nonsingular object. They apply this formula to the moduli stack of stable maps, which allows them to extend the genus zero Gromov-Witten computations of Kontsevich and others to the positive genus case. We will focus in particular on their computation of the contribution of multiple covers of a fixed curve in a Calabi-Yau 3-fold to positive genus Gromov-Witten invariants.

Kevin Jones (UIUC), An Introduction To Stable Maps and Moduli Spaces

Abstract: In recent years, physics has influenced the study of quantum cohomology. Here, we will give a low-level look at some of the basic ideas. Stable maps and moduli spaces (with their compactifications) will be discussed. We will then use moduli spaces to prove the recursive relation giving the number of degree d curves in projective 2-space passing through 3d-1 general points. A minimal understanding of projective geometry is suggested.