# Algebraic geometry and moduli seminar

## Spring Semester 2016

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
24 February 2016
13:30-15:00
Ulrike Riess
Universität Bonn
HG G 43
Abstract: In the first talk, we present the article On the Chow Ring of K3 Surfaces'' by A. Beauville and C.Voisin (2004). After introducing the necessary notation we define the canonical zero-cycle on a K3 surface and discuss some of its properties. In particular we show that the intersection product of two arbitrary one-cycles is a multiple of the canonical cycle. This proves the two dimensional case of Beauville's weak splitting property, which will be the topic of the second talk.
26 February 2016
16:00-17:15
Ulrike Riess
Universität Bonn
HG G 43
Abstract: In the second talk, we present a result on the Chow ring of irreducible symplectic varieties. The main object of interest is Beauville's conjectural weak splitting property, which predicts the injectivity of the cycle class map restricted to a certain subalgebra of the rational Chow ring (the subalgebra generated by divisor classes). For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations. After deducing that this implies the weak splitting property in many new cases, we present parts of the proof.
2 March 2016
13:30-15:00
Dr. Paolo Rossi
Université de Bourgogne
HG G 43
Abstract: In this talk I will present a series of results, applications and open problems related to the double ramification hierarchy, an integrable system associated to a cohomological field theory on the moduli space of curves which makes use of the intersection theory of the double ramification cycle. We will make contact with Gromov-Witten theory, mirror symmetry, integrable quantum field theory and deformation quantization. Most of the material is a joint work with A. Buryak.
4 March 2016
16:00-17:15
Prof. Dr. Boris Feigin
Landau Institute for Theoretical Physics
HG G 43
Abstract: We present the construction of quantum Hamiltonians which more or less automatically gives us Bethe equations. Main tool -- shuffle algebras and projection operators. We get many integrable systems -- some of them seems to be new. We discuss also the relation of this approach with the standard Q-T equation technique. The results are published (partly) in the our recent papers with Miwa, Jimbo and Mukhin.
* 7 March 2016
15:15-16:30
Dr. Ran Tessler
ETHZ/ITS
HG G 43
Abstract: This series of talks will be devoted to open GW theory. In the first talk we shall survey the main difficulties in constructing open GW theories, and describe different approaches to the subject. We then describe the full open theory of maps to a point. The last talk will be devoted to generalizations. Based on published and unpublished works, some with A. Buryak, R. Pandharipande, J. Solomon and A. Zernik.
11 March 2016
16:00-17:15
Dr. Julius Ross
Uniersity of Cambridge
HG G 43
Abstract: There are various notions of stability for sheaves (e.g. Mumford stability, Geiseker semistability) that are important in the production of projective moduli spaces. These notions usually depend non-trivially on a choice of ample line bundle, or Kahler class and different choices give rise to different, but related, moduli spaces. In this talk I will discuss joint work with Daniel Greb and Matei Toma in which we introduce a notion of Gieseker-stability that depends on several polarisations. We use this to study the change in the moduli space of Giesker semistable sheaves on manifolds giving new results in dimensions at least three, and to study the notion of Gieseker-semistability for sheaves taken with respect to an irrational Kahler class.
* 16 March 2016
12:30-14:00
Dr. Ran Tessler
ETHZ/ITS
HG G 43
Abstract: This series of talks will be devoted to open GW theory. In the first talk we shall survey the main difficulties in constructing open GW theories, and describe different approaches to the subject. We then describe the full open theory of maps to a point. The last talk will be devoted to generalizations. Based on published and unpublished works, some with A. Buryak, R. Pandharipande, J. Solomon and A. Zernik.
18 March 2016
16:00-17:15
Prof. Dr. Dragos Oprea
UC San Diego
HG G 43
Abstract: I will discuss ongoing joint work with Alina Marian and Rahul Pandharipande aimed at studying the tautological ring of the moduli space of K3 surfaces. First, I will discuss the notion of tautological. Next, I will explain a method of deriving relations between tautological classes via the geometry of the relative Quot scheme. In addition, the study of the Quot scheme of a fixed K3 surface leads to a proof in the K3 case of a conjecture of M. Lehn regarding the top Segre classes of tautological vector bundles over Hilbert schemes of points.
21 March 2016
15:30-16:45
Prof. Dr. Alexander Kuznetsov
Steklov Mathematical Institute
ITS
Abstract: A Fano fourfold is said to be of K3 type if its derived category of coherent sheaves has a semiorthogonal decomposition consisting of several exceptional objects and a derived category of a noncommutative K3 surface. Cubic fourfolds form a family of very interesting examples of this sort. They are interesting because of their intriguing birational behavior, and of their relation to hyperkahler geometry, and most probably the noncommutative K3 category is responsible for both features. I will talk about other examples of Fano fourfolds of K3 type.
23 March 2016
13:30-15:00
Dr. Ran Tessler
ETHZ/ITS
HG G 43
Abstract: This series of talks will be devoted to open GW theory. In the first talk we shall survey the main difficulties in constructing open GW theories, and describe different approaches to the subject. We then describe the full open theory of maps to a point. The last talk will be devoted to generalizations. Based on published and unpublished works, some with A. Buryak, R. Pandharipande, J. Solomon and A. Zernik.
6 April 2016
13:30-15:00
Christoph Schiessl
ETH Zurich
HG G 43
Abstract: After discussing configuration spaces and the methods to compute their cohomology, I will show some examples and present new results about the betti numbers of unordered configuration spaces of the torus.
8 April 2016
15:45-17:00
Prof. Dr. Dan Petersen
University of Copenhagen
HG G 43
Abstract: Consider a stratified topological space. If one knows the (compact support) cohomology of all open strata, then there is a well known spectral sequence calculating the (compact support) cohomology of the whole space. In this talk I describe a general construction of a dual spectral sequence, which takes as its input the cohomologies of the closed strata and computes the cohomologies of open strata. Applications are given to homological stability/representation stability theorems for configuration spaces of points.
15 April 2016
16:00-17:15
Prof. Dr. Vincent Koziarz
Université de Bordeaux
HG G 43
Abstract: I will show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on the moduli space of genus $0$ curves with $n$ marked points ${\mathcal M}_{0,n}$ are singular Kähler-Einstein metrics when ${\mathcal M}_{0,n}$ is embedded in its Deligne-Mumford-Knudsen compactification $\overline{\mathcal M}_{0,n}$. As a consequence, I will obtain a formula computing the volume of ${\mathcal M}_{0,n}$ with respect to these metrics using intersection of boundary divisors of $\overline{\mathcal M}_{0,n}$. In the case when the weights parametrizing the complex hyperbolic structures are rational, following an idea of Y.~Kawamata, I will show that the associated metrics actually represent the first Chern class of some line bundles on $\overline{\mathcal M}_{0,n}$, from which other formulas computing the same volumes will be derived.
20 April 2016
13:30-15:00
Dr. Andrea Bruno
Università di Roma III
HG G 43
Abstract: I will talk about joint work with E. Arbarello and E. Sernesi and with E. Arbarello, G. Farkas and G. Saccà. Which canonically embedded curves are hyperplane sections of K3 surfaces? In the first work, following conjectures of Wahl, Mukai and Voisin, we give a complete characterization of curves which are hyperplane sections of K3 surfaces (or of limits of such). This involves proving two conjectures stated by Wahl in 1997. In the second work we find an explicit family of curves, of any given genus, satisfying the Brill-Noether-Petri condition.
22 April 2016
16:00-17:15
Dr. Michel Van Garrel
HG G 43
Abstract: Let S be a del Pezzo surface. Local instanton numbers of S are defined (for example) by extracting integers from Gromov-Witten invariants. In the beginning of the subject, the invariants were expected to be counts of rational curves. This is however only true in low degrees. I will introduce new numbers, the log instanton numbers of S, which are weighted counts of rational curves satisfying some specified tangency conditions along an elliptic curve. I will present and motivate a conjecture that expresses the local numbers in terms of the log numbers, realising an enumerative interpretation of the former. This is joint work with Jinwon Choi and Nobuyoshi Takahashi.
27 April 2016
13:45-15:15
Dr. Hyenho Lho
KIAS
HG G 43
Abstract: I will introduce the moduli space of quasi-map introduced by Ciocan-Fontanine,Kim and Maulik. By intersection theory on the moduli space of quasi-map, we can define quasi-map invariants which are related to Gromov-Witten invariants by wall-crossing formula. In this talk, we calculate genus 1 quasi-map invariants when the target is Calabi-Yau complete intersections in projective space. Combined with wall-crossing formula, we recover genus 1 mirror theorem proved by Zinger. In the first talk, I will sketch the proof of main results and in the second talk, I will give the explicit calculations. This is joint work with Bumsig Kim.
11 May 2016
13:30-15:00
Dr. Hyenho Lho
KIAS
HG G 43
Abstract: I will introduce the moduli space of quasi-map introduced by Ciocan-Fontanine,Kim and Maulik. By intersection theory on the moduli space of quasi-map, we can define quasi-map invariants which are related to Gromov-Witten invariants by wall-crossing formula. In this talk, we calculate genus 1 quasi-map invariants when the target is Calabi-Yau complete intersections in projective space. Combined with wall-crossing formula, we recover genus 1 mirror theorem proved by Zinger. In the first talk, I will sketch the proof of main results and in the second talk, I will give the explicit calculations. This is joint work with Bumsig Kim.
18 May 2016
13:45-15:15
Dr. Margarida Melo
Roma III
HG G 43
Abstract: The universal Jacobian over curves with marked points is the moduli stack parametrizing curves with marked points together with a line bundle on the curves. By fixing a vector bundle on the universal family of the stack of stable marked curves, one can construct a compactification of the universal Jacobian stack parametrizing simple torsion-free sheaves with are semistable with respect to the given bundle. After describing the construction of such compactification, I will explain how could one apply it to study a number of different applications as the study of tautological cycles in the moduli space of curves or the construction of universal Néron models for families of curves.
27 May 2016
16:00-17:15
Dr. Daniela Egas Santander
Freie Universität Berlin
HG G 43
Abstract: In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. However, the homology of these complexes is largely unknown. In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams. I will describe how this complexes posses homological stability give some computational results.
30 May 2016
15:30-16:45
Dr. Zhiyuan Li
MPI Bonn
HG G 43
Abstract: I will talk about some recent progress about cycle theory on moduli spaces of K3 surfaces. It is proved that the rational Picard group of moduli spaces of K3 surfaces is generated by Noether-Lefschetz divisors. Built on this result, there are many applications and related questions such as the cones and the tautological classes. I will discuss our approach to these questions. This is an ongoing project joint with N. Bergeron.
17 June 2016
15:30-16:45
Dr. Nick Sheridan
Princeton
HG G 43
Abstract: I will start by explaining what mirror symmetry is about, paying special attention to the 'mirror map' which matches up the family of symplectic forms on one manifold with the family of complex structures on another. Then I will give a template for proving cases of Kontsevich's homological mirror symmetry conjecture. I will focus on one part of the template, namely a 'versality' criterion for the Fukaya category. Roughly, it gives a criterion for the existence of an appropriate mirror map (the idea is due to Seidel). I will explain how this all works for Batyrev's toric construction of mirror families from dual reflexive polytopes. The proof of homological mirror symmetry can be completed when the reflexive polytope in Batyrev's construction is a simplex: this special case of the construction is due to Greene and Plesser. The latter result is joint work with Ivan Smith.

Organizers: Rahul Pandharipande