# Tim's talks

## Research talks

### Lagrangian fibrations over quotient singularities.

October 29, 2018—Seoul (KIAS)

### On a conjecture of Matsushita (after van Geemen and Voisin).

October 12, 2018—Seoul (KIAS)

### Deformations of hyperkähler twistor spaces.

October 4, 2018—Seoul

Abstract.We obtain novel results concerning the deformation theory of twistor spaces of irreducible holomorphic symplectic (ihs) manifolds. Specifically we show that the local deformations of such twistor spaces are unobstructed. The proof is based upon an extension theorem for families of ihs manifolds which compensates for the fact that, generally, fine moduli spaces of marked ihs manifolds do not exist.

### On the local moduli of twistor spaces.

September 13, 2018—Seoul (KIAS)

Abstract.I present and discuss new results concerning the deformation theory of compact complex manifolds that are non-isotrivially fibered in irreducible holomorphic symplectic manifolds. I exemplify these results with twistor spaces.

### Deformations of hyperkähler twistor spaces.

June 28, 2018—Chemnitz

Abstract.I present some novel results concerning the deformation theory of twistor spaces of hyperkähler type. First and foremost, I show that the (local) deformations of such twistor spaces are unobstructed—a result which is new even in the K3 surface case. Time permitting, I touch upon the (global) structure of the moduli of hyperkähler twistor spaces.

### Twistor spaces of K3 surfaces.

May 16, 2018—Grenoble

Abstract.Based on a joint project with Brecan, Schwald, and Greb I present some new results on twistor spaces of hyperkähler manifolds—with an emphasis on the K3 surface case. In particular, I show that the deformations of such twistor spaces are unobstructed.

### Group actions on holomorphic Lagrangian fibrations.

March 9, 2018—Toulouse

Abstract.Let $n$ be a natural number, $G$ be a finite subgroup of the general linear group of degree $n$, and $B$ be an open neighborhood of the origin in $\mathbb C^n$ on which $G$ acts by matrix multiplication. Moreover, let $f \colon (X,\sigma) \to B$ be a holomorphic Lagrangian fibration and assume that we are given a holomorphic symplectic action of $G$ on $(X,\sigma)$ which makes the map $f$ equivariant. In my talk I ask: When the action of $G$ on $X$ is fixed point free, so that we can form the quotient $X/G$ as a complex manifold, what can we say about $G$? If $n=1$, for instance, the group $G$ is necessarily trivial. I will present partial results for $n>1$ and explain how these can be applied to study the singularities of base spaces of fibrations on irreducible holomorphic symplectic manifolds.

### Deformations of twistor spaces of K3 surfaces.

January 18, 2018—Nice

Abstract.Twistor spaces of K3 surfaces are, noticeably non-Kähler, compact complex manifolds of dimension $3$ that come equipped with a holomorphic submersion to the complex-projective line. These spaces play a fundamental role in the moduli theory of K3 surfaces. In my talk I will explain this role and review the definition/construction of twistor spaces for general hyperkähler manifolds. Then, based on joint work in progress with Ana-Maria Brecan (Bayreuth), Martin Schwald, and Daniel Greb (both Essen), I present new results concerning the deformations of twistor spaces of K3 surfaces. Moreover, I will touch upon open problems and possible research directions.

### Torelli theorems for K3 surfaces.

October 25, 2017—Torino

### Extendability of parallel sections in vector bundles.

October 12, 2017—Rauischholzhausen

### (Non-)existence of complex structures on S⁶: Etesi’s work and ideas.

March 30, 2017—Marburg

### Finite quotients of three-dimensional complex tori.

January 27, 2017—Essen

Abstract.I will report on a joint project with Patrick Graf (Bayreuth). Using Graf's results about the algebraic approximation of Kähler threefolds of Kodaira dimension zero, we show that a three-dimensional compact, connected Kähler space $X$ with canonical singularities is the finite quotient of a complex torus if and only if the first and second Chern classes of $X$ vanish, in an appropriate sense. This brings together an old theorem of Yau (where $X$ is smooth) and a theorem of Shepherd-Barron and Wilson (where $X$ is projective).

### Finite quotients of three-dimensional complex tori.

July 19, 2016—Mainz

Abstract.I will report on a current project with Patrick Graf (Bayreuth). Using Graf's recent results about the algebraic approximation of Kähler threefolds of Kodaira dimension zero, we show that a three-dimensional compact, connected Kähler space $X$ with isolated canonical singularities is the finite quotient of a complex torus if and only if the first and second Chern classes of $X$ vanish. This brings together an old theorem of Yau (where $X$ is smooth) and a theorem of Shepherd-Barron and Wilson (where $X$ is projective).

### Extendability of parallel sections in vector bundles.

January 15, 2015—Essen

### Singular irreducible symplectic spaces.

July 12, 2013—Bielefeld

### On singular symplectic complex spaces.

April 26, 2013—Bochum

### Period mappings for families of manifolds.

March 15, 2013—Freiburg

### The local Torelli theorem for irreducible symplectic spaces.

September 5, 2012—Oberwolfach

## Seminar talks

### Basics on Okounkov bodies.

Seminar program excerpt.The aim of the talk is to introduce Okounkov bodies, compute examples, and provide some background for the the material summarised by Grieve in Section 2 and 3 of his paper [Gri19]. The main reference for this should be the paper of Lazarsfeld–Mustata [LM09] and the Bourbaki-article [Bou14] of Boucksom. Try to illustrate the concepts and results with examples as much as possible. Talk to the person giving the next talk in order to be able to provide some of the preliminary results used there; i.e., it may make sense to move some material described in Talk 8 below to this talk.

May 23, 2019—Essen

### The Corlette-Donaldson Theorem.

June 30, 2016—Essen

### Zusammenhänge auf Vektorbündeln.

March 19, 2016—Bayreuth

### Stability conditions and threefold flops.

From the seminar program.In this last talk the result of Bridgeland discussed earlier will be explained using stability conditions, following [Bri06] and [Tod08].

July 16, 2015–Essen