September 29

HSE Maths department
Room 110

September 30

HSE Maths department
Room 108

Ingrid Bauer

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:

  1. $S$ is minimal of general type, or
  2. $S$ is a Del Pezzo surface of degree $\geq 5$, or
  3. $S$ 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

Paolo Cascini

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

Fabrizio Catanese

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.

Dima Kaledin

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.

Vladimir Lazić

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.

Sl̷awek Rams

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).

Sho Tanimoto

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.

Misha Verbitsky

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.