# Program

# Talks

**Ljudmila Kamenova**

**Finiteness results for hyperkaehler manifolds**

First we shall survey some classical results of Huybrechts about birational hyperkaehler manifolds. Then we'll present a proof of various finiteness results of deformations of hyperkaehler manifolds and Lagrangian fibrations.

**Yusuke Kawamoto**

**Isolated hypersurface singularities, spectral invariants, and quantum cohomology**

We discuss how the quantitative theory of Floer theory interacts with some notions of algebraic geometry such as degeneration.

**Tian-Jun Li**

**Talk 1**,

**Uniruled manifolds and rationally connected manifolds in symplectic geometry**

Uniruled manifolds and rationally connected manifolds in symplectic geometry are defined via genus 0 Gromov-Witten invariants. We discuss how they are related to the corresponding notions in algebraic geometry as well as properties of such symplectic manifolds.

**Talk 2**,

**Kodaira dimension of low dimensional symplectic manifolds**

We view the Kodaira dimension type invariant as a discrete invariant on a certain class of \(n\)-dimensional manifolds, taking values in the set {\(-\infty\), 0, 1, ..., lower floor of \(n/2\)}. The basic such invariant is of course the holomorphic Kodaira dimension for all complex manifolds. We describe such an invariant for symplectic 4-manifolds and propose a candidate for symplectic 6-manifolds.

**Talk 3**,

**Uniruled symplectic surfaces**

We survey several aspects of the geometry of uniruled symplectic surfaces.

**Mark Mclean**

**Birational Calabi-Yau manifolds have the same small quantum products**

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of symplectic cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.

**Karol Palka**

**Almost minimal models and applications**

A minimal model of a smooth projective variety of dimension at least 3 can be singular. For quasi-projective varieties and log varieties the same problem appears naturally already in dimension 2. But here, there is a modification of the Minimal Model Program based on the notion of an

*almost minimal model*, which allows to delay the appearance of singularities. For log smooth surfaces with reduced boundary it has been introduced by M. Miyanishi. We show that the idea of almost minimalization can be used more widely. When applied to a log surface \((X, rD)\), where \(r \in [0, 1]\) and \(D\) is reduced, it produces interesting contractible curves lying outside of \(D\) and meeting \(D\) in a controlled manner. We will discuss this method and some of its applications to classification problems (rational cuspidal planar curves, \(\mathbb{Q}\)-acyclic surfaces, singular log del Pezzo surfaces of rank 1 in arbitrary characteristic).

**Deepam Patel**

**Local monodromy of constructible sheaves**

Let \(X\) be a complex algebraic variety, and \(f: X \to D\) a proper morphism to a small disk which is smooth away from the origin. In this setting, the higher direct images of the constant sheaf form a local system on the punctured disk, and the Local Monodromy Theorem (due to Brieskorn-Grothendieck-Griffiths-Landsman) asserts that the eigenvalues of local monodromy are roots of unity. In this talk, we will discuss generalizations of this result to the setting of arbitrary morphisms between complex algebraic varieties, and with coefficients in arbitrary constructible sheaves. If there is time, I'll discuss applications to variation of monodromy in abelian covers, and applications to the monodromy of alexander modules.

This is based on joint work with Madhav Nori.

**Martin Pinsonnault**

**Embeddings of symplectic balls in \(\mathbb{CP}^2\) and configuration spaces**

Existence of symplectic embeddings of \(k\) disjoint balls of given capacity \(c_1,\ldots, c_k\) into a given symplectic manifold \((M,\omega)\) is a central problem in symplectic topology. However, beside a few examples, very little is known about the space of all such embeddings. In this talk, I will discuss the case of rational \(4\)-manifolds of small Euler numbers, with a special attention to the minimal manifolds \(\mathbb{CP}^2\) and \(\mathbb{S}^2 \times \mathbb{S}^2\). For rational manifolds, a very rich and intricate picture emerges that blends symplectic topology, complex geometry, and algebraic topology.

**Eric Riedl**

**Geometric Lang Vojta for hypersurfaces of degree \(2n\) in \(\mathbb{P}^n\)**

Inspired by conjectures from number theory, the Geometric Lang-Vojta Conjecture predicts that given a divisor \(D\) in a variety \(X\), if \(K_X + D\) is big then the complement of \(D\) in \(X\) is algebraically hyperbolic outside of some exceptional set \(S\) in \(X\). Being algebraically hyperbolic means that any curve \(C\) in \(X\) of high degree will have the Euler characteristic of \(C\) minus \(D\) is large. In other words, for high degree curves, either the genus is large or the number of times it meets \(D\) is large. We prove this Conjecture for \(D\) a very general hypersurface in \(\mathbb{P}^n\) of degree \(2n\). The case of very general quartics in \(\mathbb{P}^2\) had been of interest for decades, and in this case, we completely characterize the exceptional set and give a sharp bound on the Euler characteristic in terms of the degree. This is joint with Xi Chen and Wern Yeong.