Complex Geometry(22 Jul - 9 Aug 2013)
## ~ Abstracts ~
Florin Ambro, Romanian Academy of Sciences, RomaniaThe 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.
Caucher Birkar, University of Cambridge, UKI 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.
Paolo Cascini, Imperial College, UKI will discuss some recent progress towards the base point free theorem in positive characteristic. Joint work with H. Tanaka and C. Xu.
Fabrizio Catanese, Universität Bayreuth, GermanyFor 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.
Ivan Cheltsov, University of Edinburgh, UKThis 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.
Xi Chen, University of Alberta, CanadaIt 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.
Meng Chen, Fudan University, ChinaWe present some examples of 3-folds of general type whose can maps are of fibre type.
Izzet Coskun, University of Illinois at Chicago, USAIn 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.
Tommaso de Fernex, University of Utah, USAThe 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.
Lawrence Ein, University of Illinois at Chicago, USAWe'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.
Philippe Eyssidieux, Université Grenoble I, FranceWe will survey our joint work with Campana and Claudon, arxiv:1302.5016
Baohua Fu, Chinese Academy of Sciences, ChinaWe shall show that among Fano manifolds of Picard number 1 with smooth VMRT, projective space is the only one compactifying C ^{n} equivariantly in more than one ways. This is a joint work with Jun-Muk Hwang.
Yoshinori Gongyo, University of Tokyo, JapanI 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.
Jun-Muk Hwang, Korea Institute for Advanced Study, KoreaFor 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.
Shin-Yao Jow, National Tsing Hua University, TaiwanWe determine the effective cone of the Quot scheme parametrizing all rank r, degree d quotient sheaves of the trivial bundle of rank n on P^{1}. 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.
Masayuki Kawakita, Kyoto University, JapanI shall discuss a connectedness lemma on the spectrum of a formal power series ring, and its possible applications.
Jonghae Keum, Korea Institute for Advanced Study, KoreaIt 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 $p\neq 2,3$.
Steven Lu, Université du Québec Montréal, CanadaWe 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.
Mircea Mustata, University of Michigan, USAI 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.
Sui Chung Ng, Temple University, USAThe 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.
Keiji Oguiso, Osaka University, JapanWe 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.
Mihai Paun, Korea Institute for Advanced Study, KoreaWe will present a few results in connection with the celebrated generic semi-positivity theorem of Y. Miyaoka, together with some applications.
Yum-Tong Siu, Harvard University, USAWill 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.
Xiaotao Sun, Chinese Academy of Sciences, ChinaI 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.
Shengli Tan, East China Normal University, ChinaI 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.
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.
Fei Ye, The University of Hong Kong, Hong KongA 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.
Sai-Kee Yeung, Purdue University, USAThe 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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