Titles and Abstracts

Yonghwa Cho_Korea Institute for Advanced Study

Exceptional collections arising from toric degenerations of del Pezzo surfaces

By the work of Hacking and Prokhorov, $\mathbb{Q}$-Gorenstein degenerations of del Pezzo surfaces to toric varieties with Picard number 1 are classified by Markov type equations. Precisely the same Markov type equations appear when one considers three block collections in del Pezzo surfaces. In this talk, we first discuss the relation between these degenerations and three block collections. Afterwards, we apply relative MMP to produce further examples, for instance, exceptional collections associated with the toric degenerations to higher Picard numbers. This talk is based on the work in progress.

Hakho Choi_Korea Institute for Advanced Study

Symplectic fillings of weighted homogeneous complex surface singularities

One of the fundamental problems in symplectic 4-manifold topology is in classifying symplectic fillings of certain 3-manifolds equipped with a natural contact structure. If we get the classification result, then it is natural to ask that Is there any surgery description of those fillings. In this series of talks, we discuss classification of minimal symplectic fillings of weighted homogeneous surface singularities satisfying certain conditions. Furthermore, we demonstrate that every minimal symplectic filling of weighted homogeneous complex surface singularities satisfying the conditions can be obtained by a sequence of rational blowdowns from the minimal resolution. This is joint work with Jongil Park.

Jonathan David Evans_Lancaster University

Lectures on symplectic embeddings of rational homology balls

Suppose a smooth complex projective surface $X$ degenerates and develops a Wahl singularity. From the viewpoint of symplectic topology what is happening is that a certain Lagrangian subset in $X$ (the "vanishing cycle") is collapsing to a point in the singular surface. This vanishing cycle is called a "Lagrangian pinwheel". A neighbourhood of a Lagrangian pinwheel is a symplectically embedded rational homology ball in $X$. One can therefore hope to understand the Wahl singularities which can develop by studying the symplectically embedded rational homology balls in $X$. In joint work with Ivan Smith, we used this idea to prove bounds on the length of Wahl singularities which can occur for surfaces of general type (results which were independently obtained by Urzúa and Rana using purely algebro-geometric methods). In the symplectic world, the general type condition is replaced by the "negative monotonicity condition" that the cohomology class of the symplectic form equals the canonical class. If one drops the negative monotonicity condition, Giancarlo Urzúa and I were able to construct infinite families of symplectically embedded rational homology balls which violate the bounds which hold in the general type case. This uses ideas from the work of Hacking-Tevelev-Urzúa on flips, reinterpreted in terms of symplectic geometry and almost toric fibrations. As a byproduct of this, we are able to give an interpretation of the flip in terms of mutations of almost toric fibrations. In these lectures, I will explain some of these ideas.

Wenfei Liu_Xiamen University

On the volumes of log canonical surfaces

The geography of singular surfaces acquires much more complexity due to their volumes, which are only rational numbers rather than integers. By a fundamental result of Alexeev, solving a conjecture by Kollár, the set of volumes of all projective log canonical surfaces satisfies the so-called descending chain condition (DCC). This has far-reaching consequences such as the boundedness of the moduli spaces of projective log canonical surfaces. In the first talk I will introduce the basics about log canonical surfaces such as their singularities and volumes. I will state Alexeev's DCC result and explain how this was achieved as well as its relation to boundedness of moduli spaces. In the second talk I will talk about recent progress concerning the geography of projective log canonical surfaces. The focus is on accumulation points and the minima of the volume sets for different classes of log canonical surfaces.

Kyungbae Park_Seoul National University

1. Definite intersection forms of 4-manifolds with boundary

The intersection pairing of the second homology group of a compact oriented 4-manifold is a symmetric bilinear form on an integer lattice, called the intersection form. In this talk, we discuss which definite forms can be realized as the intersection forms of 4-manifolds bounded by a given 3-manifold. In particular, we introduce some conditions on such form induced from the Donaldson’s diagonalization theorem and Heegaard Floer theory.

2. Spherical 3-manifolds bounding rational homology balls

In this talk, we give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. This is joint work with Dong Heon Choe.

İrem Portakal_Otto-von-Guericke-Universität

On deformations and mutations of toric varieties

In the first chapter we start by introducing the deformation theory of toric varieties using combinatorial tools from toric geometry. The theory was first developed by Altmann and these methods have been recently generalized to the case of T-varieties. We take a short tour around these results and bring our attention to toric Fano varieties. One of the questions arising from the minimal model program is the classification of Fano varieties. Akhtar, Coates, Galkin, and Kasprzyk introduced the notion of mutations which is a special class of birational transformation of Fano varieties. In the toric case, it turns out that the mutation of a toric Fano variety $TV(P)$ can be understood by a piecewise-linear transformation, i.e. a combinatorial mutation of its associated dual polytope $P*$. In the second chapter we first present certain concrete applications of the methods which were introduced in the first chapter. Ilten showed that for a given mutation, there exists a flat projective family over the projective line such that zero fiber is $TV(P)$ and the infinity fiber is the toric variety associated to the dual polytope of the combinatorial mutation of $P*$. Motivated by this result, we compare mutations and deformations in the case of $\mathbb{Q}$-Gorenstein smoothings of weighted projective planes with weights giving solutions to Markov equation.

Giancarlo Urzúa_Pontificia Universidad Católica de Chile

On the explicit MMP for one parameter degenerations of surfaces

These three lectures will be on $T$-singularities and MMP for degenerations of surfaces. I will mainly focus on describing a toolkit to work with it, the general picture and several explicit examples. Open questions will be discussed.

Weiyi Zhang_University of Warwick

Pseudoholomorphic subvarieties in an exceptional curve class

Let J be an almost complex structure tamed by a symplectic form on a 4-manifold. I will discuss J-holomorphic curves in an exceptional curve class, in particular the (non-)uniqueness, and how it is related to bounds on Wahl singularities. I will start with an overview of some useful techniques in degenerations of pseudoholomorphic curves, including the cone theorem and J-nef classes in almost complex 4-manifolds.