Complex Geometry - IMS

Complex Geometry
(22 Jul - 9 Aug 2013)

~ Abstracts ~

An injectivity theorem
Florin Ambro, Romanian Academy of Sciences, Romania

The injectivity theorem of Esnault and Viehweg is a basic result which implies all known vanishing and extension theorems used in the log minimal model program. I will present the generalization of their result to the log canonical case, and some applications to new extension and vanishing theorems.

Manifolds with semi-ample anti-canonical bundle
Caucher Birkar, University of Cambridge, UK

I will discuss properties of images of a manifold with semi-ample anti-canonical bundle under surjective morphisms. It turns out that the image also has semi-ample anti-canonical bundle if the morphism is smooth.

Base point freeness in positive characteristic
Paolo Cascini, Imperial College, UK

I will discuss some recent progress towards the base point free theorem in positive characteristic. Joint work with H. Tanaka and C. Xu.

Moduli spaces of automorphism marked varieties: the case of curves and surfaces
Fabrizio Catanese, Universität Bayreuth, Germany

For several reasons it is interesting to consider moduli spaces of triples (X,G,a) where X is a projective variety, G is a finite group, and a is an effective action G on X. If X is the canonical model of a variety of general type, then G is acting linearly on some pluricanonical model, and we have a moduli space which is a finite covering of a closed subspace M^G of the moduli space.

In the case of curves this investigation is related to the description of the singular locus of the moduli space M_g, for instance of its irreducible components (due to Cornalba), and of its compactification M_g. In the case of surfaces there is another occurrence of Murphy´s law, as shown in my joint work with Ingrid Bauer: the deformation equivalence for minimal models S and for canonical models differs drastically (nodal Burniat surfaces being the easiest example).

In the case of curves, there are interesting relations with topology. Moduli spaces of curves with a group G of automorphisms of a fixed topological type have a description by Teichmueller theory, which naturally leads to conjecture genus stabilization for rational homology groups.

I will then describe two equivalent description of its irreducible components, surveying known irreducibility results for some special groups. A new fine homological invariant was introduced in my joint work with Loenne and Perroni: it allows to prove genus stabilization in the ramified case, extending a theorem of Dunfield and Thurston in the easier unramified case.

Affine cones over del Pezzo surfaces
Ivan Cheltsov, University of Edinburgh, UK

This talk is about joint paper with Park and Won (see http://arxiv.org/abs/1303.2648). Our main result is about smooth cubic surfaces. We answer the old question by Misha Zaidenberg and Hubert Flenner. Namely, we prove that affine cones over smooth cubic surfaces do not admit non-trivial actions of the additive group. The proof involves functional analogue of the alpha-invariant of Tian and some classical constructions that go back to Manin and Serge.

Density of rational curves on K3 surfaces
Xi Chen, University of Alberta, Canada

It was conjectured that the union of rational curves on a K3 surface is dense in the strong topology. We gave a proof of this conjecture for a general K3 surface. This is a joint work with James D. Lewis.

Some examples of 3-folds canonically fibred by curves and surfaces
Meng Chen, Fudan University, China

We present some examples of 3-folds of general type whose can maps are of fibre type.

The birational geometry of the Hilbert scheme of points on surfaces and Bridgeland stability
Izzet Coskun, University of Illinois at Chicago, USA

In this talk, I will report on joint work with D Arcara, A Bertram and J Huizenga on the birational geometry of the Hilbert scheme of points on surfaces. The notion of Bridgeland stability gives a new way to study the geometry of the moduli spaces of sheaves on surfaces. I will discuss recent progress in understanding the cones of ample and effective divisors and the stable base locus decomposition of the effective cone, concentrating on the example of Hilbert schemes of points on P^2. I will describe a correspondence between the Mori walls in the Neron-Severi space and the Bridgeland walls in the stability manifold.

Birational rigidity of Fano hypersurfaces of index one
Tommaso de Fernex, University of Utah, USA

The study of the birational group of Fano hypersurfaces was undertaken by Fano in dimension three, and has lead to the proof, by Iskovskikh and Manin, that smooth quartic threefolds are not rational. Their work eventually led to the notion of birational rigidity, and higher dimensional Fano hypersurfaces of index one have since been investigated from this point of view. In this talk I will overview the problem and the latest results in this direction.

Stability of syzyby bundles
Lawrence Ein, University of Illinois at Chicago, USA

We'll discuss joint work with Lazarsfeld and Mustopa. We show under various assumptions that if L is a sufficiently ample line bundle on a smooth projective variety, then the syzygy of L is stable.

Linear representations of Kähler groups and uniformization of compact Kähler manifolds
Philippe Eyssidieux, Université Grenoble I, France

We will survey our joint work with Campana and Claudon, arxiv:1302.5016

On equivariant compactifications of Cⁿ
Baohua Fu, Chinese Academy of Sciences, China

We shall show that among Fano manifolds of Picard number 1 with smooth VMRT, projective space is the only one compactifying Cⁿ equivariantly in more than one ways. This is a joint work with Jun-Muk Hwang.

Surface of globally F-split type
Yoshinori Gongyo, University of Tokyo, Japan

I will explain about surface of globally F-regular and F- split type. In particular we show these are log Fano and log Calabi-- Yau respectively. This is a joint work with Shunsuke Takagi.

Isotrivial VMRT-structures
Jun-Muk Hwang, Korea Institute for Advanced Study, Korea

For a uniruled projective manifold X, the VMRT's at general points of X are a family of subvarieties in the projectivized tangent spaces of X determined by the tangent directions to minimal rational curves on X. The study of this family of subvarieties involves both algebraic geometry and differential geometry. To highlight the differential geometric component, we investigate the case when it is isotrivial as a family of projective varieties, namely, when all members are isomorphic to a fixed projective variety Z. We will formulate conditions on Z under which the family of VMRT's becomes locally flat in the sense of Cartanian geometry. These conditions can be verified when Z is a complete intersection, excluding some exceptional degrees.

The effective cone of the space of maps from P¹ to a Grassmannian
Shin-Yao Jow, National Tsing Hua University, Taiwan

We determine the effective cone of the Quot scheme parametrizing all rank r, degree d quotient sheaves of the trivial bundle of rank n on P¹. More specifically, we explicitly construct two effective divisors which span the effective cone, and we also express their classes in the Picard group in terms of a known basis.

A connectedness lemma on the spectrum of a formal power series ring
Masayuki Kawakita, Kyoto University, Japan

I shall discuss a connectedness lemma on the spectrum of a formal power series ring, and its possible applications.

Orders of automorphisms of K3 surfaces
Jonghae Keum, Korea Institute for Advanced Study, Korea

It is a natural and fundamental problem to determine all possible orders of automorphisms of K3 surfaces in any characteristic. Even in the case of complex K3 surfaces, this problem has been settled only for symplectic automorphisms and purely non-symplectic automorphisms (Nikulin, Kond\={o}, Oguiso, Machida-Oguiso).

In a recent work I solve the problem in all characteristics except 2 and 3. In particular, 66 is the maximum possible finite order in each characteristic $p\neq 2,3$.

Seminegativity of the log tangent sheaf and the log canonical divisor
Steven Lu, Université du Québec Montréal, Canada

We will discuss with applications some natural types of seminegativity of the log tangent bundle connected with complex pseudo hyperbolicity and their relationships with the canonical divisor:

1st type: seminegative holomorphic sectional curvature. In this case, the canonical is nef and the maximal rank of subspaces of the tangent spaces in which the sectional curvature is strictly negative bounds the numerical kodaira dimension. This is joint work with Gordon Heier and Bun Wong.

2nd type: Absence of nonconstant hol/alg maps from C (Brody/Mori hyperbolicity). In this case the canonical is ample/nef and we discuss the log generalizations. This is joint work with De-Qi Zhang.

3rd type (if time permits): Miyaoka's generic seminagativity of the log tangent sheaf and its very well known proof without the use of characteristic p>0 technique in the case of pseudoeffective canonical, recently written by Campana-Paun. This is a report on this work of Campana-Paun and of my student: Behrouz Taji.

Numerical dimension of divisors in positive characteristic
Mircea Mustata, University of Michigan, USA

I will discuss a positive characteristic version of a result of Nakayama concerning the numerical dimension of pseudo-effective divisors. The proof relies on the trace map for the Frobenius morphism. This is joint work with Paolo Cascini, Christopher Hacon, and Karl Schwede.

Rigidity of proper holomorphic mappings among classical domains
Sui Chung Ng, Temple University, USA

The study of proper holomorphic mappings is a classical subject in Several Complex Variables. In this talk, we will look at the cases for classical symmetric domains on Grassmannians. These include in particular Type-I bounded symmetric domains and generalized balls. Some interesting linkage between the proper mapping problems among different types of domains will be explained.

Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy
Keiji Oguiso, Osaka University, Japan

We present the first explicit examples of a rational threefold and a Calabi-Yau threefold, admitting biregular automorphisms of positive entropy not preserving any dominant rational maps to lower positive dimensional varieties. The most crucial part is the rationality of the quotient threefold of a 3-dimensional torus of product type. This is a joint work with Doctor Tuyen Trung Truong.

Metrics with cone singularities and generic semi-positivity
Mihai Paun, Korea Institute for Advanced Study, Korea

We will present a few results in connection with the celebrated generic semi-positivity theorem of Y. Miyaoka, together with some applications.

Very ampleness part of Fujita's conjecture
Yum-Tong Siu, Harvard University, USA

Will talk about the proof that there is a positive integer a(n) which is explicitly expressible in terms of n such that mL+K is very ample for any positive line bundle L on a compact conplex manifold X of complex dimension n and for any integer m no less than a(n), where K is the canonical line bundle of X.

Revisit factorizations of generalized theta functions
Xiaotao Sun, Chinese Academy of Sciences, China

I proved (in JAG 2000) a factorization theorem of generalized theta functions and vanishing theorems on moduli spaces of semistable parabolic sheaves on irreducible curves. In this talk, I will show a vanishing theorem on moduli spaces of semistable parabolic bundles on a reducible curve, which together with a factorization theorem on reducible curves (I proved in Ark. Mat. 2003) give a pure algebraic geometry proof of Verlinde formula.

Invariants of families of curves and abelian automorphisms groups of surfaces of general type
Shengli Tan, East China Normal University, China

I am going to talk about our recent work on the invariants of some special families of algebraic curves and the optimal upper bounds of the abelian automorphisms groups of surfaces of general type.

Nonexistence of asymptotical GIT compactification
Chenyang Xu, Beijing University, China

(Joint with Xiaowei Wang) By comparing different stability notions and related invariants, we prove that there exists a family of canonically polarized manifolds, e.g., hypersurfaces in P^3, which doesn't have asymptotical GIT limit.

Classifications of arrangements of 10 projective lines
Fei Ye, The University of Hong Kong, Hong Kong

A line arrangement $\mathcal{A}$ is a finite collection of projective lines. A central topic of line arrangements is to study geometry of the complement of $\mathcal{A}$ and how it interacts with the combinatorial structure. It is well-known that the diffeomorphic type of the complement of a line arrangement $\mathcal{A}$ is rigid if the moduli space $\mathcal{M}_{\mathcal{A}}$ is connected. That motivates us to classify moduli spaces of line arrangements with respect to irreducibility.

In this talk, I will present a complete list of reducible moduli spaces of arrangements of 10 lines and explain ideas of the classification. This is joint work with Meirav Amram, Moshe Cohen and Mina Teicher.

Kobayashi hyperbolicity and Finsler metrics on some family of complex manifolds
Sai-Kee Yeung, Purdue University, USA

The purpose of the talk is to present some joint work with Wing-Keung To on hyperbolicity of some families of complex manifolds, generalizing results on dimension one fibers to higher dimensional ones.

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