HSE Maths department

*Room 110*

HSE Maths department

*Room 108*

**Celebration**

**Title:** On rigid compact complex surfaces and manifolds

**Abstract:** A compact complex manifold $X$ is rigid if it has no nontrivial deformations.
The only rigid complex curve is the projective line; for dimension 2 we prove:

*Theorem.* Let $S$ be a compact complex surface, which is rigid, then:

- $S$ is minimal of general type, or
- $S$ is a Del Pezzo surface of degree $\geq 5$, or
- $S$ is an Inoue surface.

We explain different concepts of rigidity, their relations and give new examples and pose open questions.

This is joint work with F. Catanese

**Title:** Effective birationality for Fano varieties

**Abstract:** I will explain some recent results on the anti-pluricanonical morphism for singular Fano varieties.
Joint work with J. McKernan

**Title:** Rigid Manifolds, Projective classifying spaces, Inoue type varieties and deformation to hypersurface embeddings

**Abstract:**
A classifying space is a space whose universal covering is contractible. There are not so many projective varieties which are
PCS = projective classifying spaces, but many rigid surfaces of general type tend to be PCS. I shall discuss some joint work with
Ingrid Bauer on rigid manifolds, and some surfaces for which it is still open the question whether they are rigid, respectively PCS.
I shall then explain some examples how hyper surfaces in PCS lead to varieties whose moduli spaces can be understood through topology,
as the Inoue type varieties, which are quotients of hyper surfaces in PCS. Strong rigidity of these depend on strong conditions, and
one would like to enlarge the definition in the case of high degree. Studying the deformations of these, one is naturally lead to study
finite maps $f$ which deform to hypersurface embeddings. After giving some examples, I shall explain some work in progress together
with Yongnam Lee, characterising the case where the map $f$ has prime degree.

**Title:** Witt vectors, commutative and non-commutative

**Abstract:** Witt vectors $W(A)$ of a commutative ring $A$ were discovered 80
years ago, but they still pop up in unexpected places and are the subject
of continuous research. One question that has been solved only recently is
how to generalize Witt vectors to the case when $A$ is not commutative. I
am going to review the classical theory, and then show how a very natural
modification leads to the non-commutative case.

**Title:** On two conjectures

**Abstract:** I will discuss recent progress on two conjectures in birational geometry: the nonvanishing conjecture and a conjecture of Mumford. This is joint work with Thomas Peternell.

**Title:** On lines on surfaces

**Abstract:** I will present various results on lines on smooth and non-smooth low-degree surfaces in $\mathbb{P}^3(K)$. In particular I will introduce b-functions that can be applied to count lines along a line and find surfaces with lines (partially based on joint project with M. Schuett (LUH).

**Title:** On the geometry of exceptional sets in Manin's conjecture

**Abstract:** Manin's conjecture is a conjectural asymptotic formula for the counting function of rational points on a Fano variety, and it predicts an explicit asymptotic formula in terms of geometric invariants of the underlying variety after removing an exceptional set. In this talk I would like to discuss the geometry of this exceptional set using birational geometry, e.g., the minimal model program. This is joint work with Brian Lehmann.

**Title:** Lagrangian fibrations and complex curves

**
Abstract:
**
Consider a complex-valued skew-symmetric 2-form
on a $4n$-dimensional real vector space $V$. Assume that
real part of $w$ is non-degenerate, and $w^{n+1}=0$.
Then $(V,w)$ is called "complex symplectic vector space".
It admits a unique complex structure operator such that
$w$ is a complex linear symplectic form.

Let $w$ be a complex-valued differential form on a manifold $M$ giving a structure of complex symplectic vector space to tangent spaces at all points of $M$. Then $M$ admits an almost complex structure, which is integrable whenever $w$ is closed. In this case, $w$ is holomorphically symplectic.

Let now $(M,w)$ be a holomorphically symplectic manifold, $p: M \to B$ a Lagrangian fibration, and $u$ a closed $(1,1)$-form on $B$. Then $w + t p^*(u)$ defines a complex symplectic structure for all $t$ in $C$. The corresponding deformation family is called "degenerate twistor deformation".

Consider now a complex curve $S$ in $B$, and let $S' \subset M$ be a smooth curve which projects to $S$ diffeomorphically. We prove that there exists a degenerate twistor deformation $M'$ of $M$ such that $S'$ is holomorphic in $M'$. This gives a topological construction of holomorphic curves in deformations of hyperkahler manifolds. This is a joint work with Fedor Bogomolov and Rodion Deev.