Classification of Algebraic Varieties and related topics

8-15 settembre 2013, Grand Hotel San Michele, Cetraro, Italy

.. Participants Abstracts Schedule How to get to the conference Special advices and practical informations

Abstracts ....  Minimal Model Program on quasi polarized variety Marco Andreatta (Università di Trento) Abstract ....  ....  Inoue type manifolds and new surfaces with χ = 1 Ingrid Bauer (Universität Bayreuth) Inoue type manifolds are projective manifolds which are fnite étale Galois quotients of an ample divisor in a product of Abelian varieties and a product of curves. Under certain hypotheses any manifold homotopically equivalent to an Inoue type variety is also an Inoue type variety. This fact has important applications to the theory of moduli of surfaces of general type, because it often allows to conclude that a given open subset of the moduli space is in fact a connected component. We give some applications to the construction and determination of the subset of the moduli space of some new surfaces of general type with minimal Euler-Poincaré characteristic. This is joint work with F. Catanese and D. Frapporti. ....  ....  On the Hodge theoretic approach to the irrationality problem for cubic fourfolds Christian Böhning (Universität Hamburg) We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold. We also discuss some possibilities to modify this approach (in particular in the light of Griffiths' theory of infinitesimal variation of Hodge structure). This is joint work with Asher Auel and Hans-Christian von Bothmer. ....  ....  On the categorical approach to the irrationality problem for cubic 4-folds Hans Christian Graf von Bothmer (Universität Hamburg) ....  ....  Birational stability of the orbifold cotangent bundle (j.work with M. Paun) Frédéric Campana (Université Nancy) If (X,D) is a complex projective log-canonical pair, we show that its "orbifold" cotangent bundle-to be defined- is generically semi-positive if its canonical bundle is pseudo-effective. This extends to the "orbifold" frame, but with a different proof, the similar classical result of Y. Miyaoka. Combined with [BCHM], this implies that the canonical bundle of (X,D) is big if some tensor power of its (orbifold) cotangent bundle contains a big line bundle. A fundamental result of Viehweg-Zuo then shows that the (quasi-projective) base of an algebraic family of canonically polarised manifolds is of log-general type if it has "maximal variation" of its moduli. ....  ....  Prime divisors and birational geometry in Fano manifolds Cinzia Casagrande (Università di Torino) Let X be a smooth, complex Fano variety, D a prime divisor in X, and set c(D):=dim ker(r: H2(X,R) →H2(D,R)), where r is the natural restriction map. It is a special property of Fano manifolds that the presence of a prime divisor D with large c(D) has consequences on the geometry of X. More precisely, we define: cX:=max{c(D)|D is a prime divisor in X}. Then cX≤ 8, and if cX is at least 2, then we get some special properties of X. We will explain this result, which relies on a construction in birational geometry; then we will focus on the case cX=2, which is new. ....  ....  Abelian varieties in Brill-Noether loci and applications to irregular surfaces Ciro Ciliberto (Università di Roma "Tor Vergata") I will talk about recent results in collaboration with M. Mendes Lopes and R. Pardini. First, the classification of curves such that some Brill--Noether locus Wds (strictly contained in the Jacobian J of the curve), contains in turn a subvariety stable under the action of a non--zero abelian sub--variety of J and is of maximal dimension under this condition (this extends previous results by Abramovich--Harris and Debarre-Fahaloui). Then I will apply this to the classification of minimal irregular surfaces such that K2=2pg (a result of Debarre says that K2 ≥ 2pg forn any minimal irregular surface). ....  ....  Mirror symmetry and Fano manifolds Alessio Corti (Imperial college, London) I discuss work in progress with Coates, Galkin, Golyshev and Kasprzyk. We show by explicit computation that the 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a combinatorially defined collection of Laurent polynomials called Minkowski polynomials. This strongly suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors. The advantage of this approach is that the mirrors are objects of combinatorics: given enough computing power their classification is a trivial matter. I will sketch a possible way to construct Fano manifolds from the mirrors by using the Gross--Siebert program, smoothing degenerate Fano varieties made of toric components glued torically. ....  ....  Algebraic hyperbolicity of certain affine surfaces Pietro Corvaja (Università di Udine) Let $X$ be a smooth affine surface; it can be compactified via a smooth complete surface $\tilde{X}\supset X$ such that the complement $D:=\tilde{X}\setminus X$ is a union of smooth curves intersecting transversely. Let $K$ be a canonical divisor of $\tilde{X}$. Whenever the sum $K+D$ is a big divisor, the following bound is conjectured: there exists a positive number $c>0$ such that for every curve ${\cal C}\subset X$, \begin{equation*} \deg({\cal C})\leq c\cdot\max(1,\chi_{\cal C}), \end{equation*} where the degree $\deg(\cdot)$ refers to a given projective embedding of $\tilde{X}$ and $\chi_{\cal C}$ denotes the Euler characteristic of the curve ${\cal C}$. This conjecture is a special case of the function-field version of the Lang-Vojta conjectures. In a joint work with U. Zannier, we prove such a bound whenever the surface $X$ is a ramified cover of the torus ${\Bbb G}_m^2$. This in particular applies to the complement in the plane of a three component curve with normal crossing singularities and degree at least 4 . ....  ....  Quadratic line complexes Olivier Debarre (ENS, Paris) In this talk, a quadratic line complex is the intersection, in its Pluecker embedding, of the Grassmannian of lines in an 4-dimensional projective space with a quadric. We study the moduli space of these Fano 5-folds in relation with that of EPW sextics. ....  ....  Mok characteristic varieties and the normal holonomy group Antonio Josè Di Scala (Politecnico di Torino) Let $M$ be a (connected) complex manifold holomorphically immersed in $\mathbb{CP}^n$. Let $T\mathbb{CP}^n|_M= TM \oplus \nu(M)$ be the orthogonal splitting in tangent and normal bundles with respect to the Fubini-Study metric. The normal bundle is endowed with a canonical connection $\nabla^\perp$ called the normal connection. The normal holonomy group is by definition the holonomy group of the normal connection of the normal bundle. We will explain how normal holonomy groups are related to the so called Mok characteristic varieties associated to bounded symmetric domains. Joint work F. Vittone ....  ....  An infinitesimal Torelli theorem for projective hypersurfaces with simple singualrities. Philippe Eyssidieux (Université de Grenoble) Based on my joint work with Damien Mégy arXiv:1305.3780. A construction of Carlson-Toledo gives an orbifold on a big open subset of the projective space of degree d projective hypersurface whose orbifold locus is the discriminant carrying a natural VHS extending the one given by monodromy on the smooth locus. I will survey its properties which are interesting from the perspective of uniformization of projective manifolds and describe the proof of an infinitesimal Torelli theorem along the natural strata. ....  ....  Totally geodesic submanifolds in the Jacobian locus Paola Frediani (Università di Pavia) Let J be the Jacobian locus, A the moduli space of principally polarized abelian varieties and T be the closure of J in A. Using results obtained by Colombo, Pirola, Tortora and myself on the second fundamental form of J with respect to the Siegel metric, we look for obstructions to the existence of totally geodesic submanifolds of A which are contained in T and intersect J. According to the Coleman-Oort conjecture, for sufficiently high genus such submanifolds should not exist under an additional assumption of arithmetic nature. In this seminar we will give an upper bound on the dimension of these totally geodesic submanifolds which only depends on the genus. Then we will also study particular families obtained as cyclic coverings of P^1. This is a joint work with Elisabetta Colombo and Alessandro Ghigi. ....  ....  Examples of homologically trivial rational curves on certain Moishezon twistor spaces Akira Fujiki (Osaka University) There exist many Moishezon twistor spaces, which are almost never projective by a theorem of Hitchin, It is then interesting to know the way how they are non-projective. In this talk we exihibit explicit examples of homologically trivial (reducible) rational curves for the twistor spaces associated to the self-dual metrics due to LeBrun and Joyce, which are defined on any number of connected sums of of copies of complex projective plane. These twistor spaces are charactereized by the existence of non-trivial group actions. We specify a special configuration of curves on each of these twistor spaces and apply to them a previously obtained general criterion for the homological triviality of certain configuration of rational curves on a Moishezon manifold with C^*-actions. ....  ....  On characterizations of log Fano varieties Yoshinori Gongyo (University of Tokyo and Imperial college) I will talk about two characterizations of log Fano varieties by using the Cox rings and anti-canonical rings. ....  ....  Structure theory of singular varieties with trivial canonical class Daniel Greb(Universität Bochum) I will discuss recent work of Stefan Kebekus, Thomas Peternell, and myself concerning the structure theory of singular varieties with trivial canonical class. I will present an infinitesimal version of the classical decomposition theorem of Beauville, Bogomolov, Fujiki, and others, discuss etale fundamental groups of varieties with klt singularities, and use these ingredients to prove a characterisation of singular varieties that arise as finite group quotients of abelian varieties. ....  ....  Perfect forms, matroids and the cohomology of compactifications of Ag Klaus Hulek (Leibniz Universität Hannover) In this talk we will discuss the cohomology of (partial) compactifications of Ag. We will show that the matroidal (partial) compactification and the first Voronoi or perfect cone compactification have stable cohomology, thus extending results of Borel and Charney and Lee. We will also show that the stable cohomology is generated by algebraic classes and will describe concrete generators in small degree. This is joint work with S. Grushevsky and O. Tommasi. ....  ....  Equivariant compactifications of vector groups Jun-Muk Hwang (Korean Institute for Advance Studies) In a joint work with Baohua Fu, we study how many different ways a Fano manifold of Picard number 1 can compactify the complex vector group equivariantly. Hassett and Tschinkel showed that projective space of dimension at least 2 can be realized as equivariant compactifications of the vector group in many distinct ways. Our result says that projective space is an exception: among Fano manifolds of Picard number 1 with smooth VMRT, projective space is the only one compactifying the vector group equivariantly in more than one ways. ....  ....  Relations between some invariants of algebraic varieties in positive characteristic Toshiyuki Katsura (Hosei University) We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. As an example, we calculate the a-number for Fermat surfaces. ....  ....  On tilting generators Yujiro Kawamata (University of Tokyo) I will consider tilting generators of derived categories for singular varieties such as weighted projective spaces. ....  ....  Certain vanishing on Fake Projective Planes Jonghae Keum (KIAS, Seoul) ....  ....  On certain duality of Néron-Severi lattices of supersingular K3 surfaces (joint work with Ichiro Shimada) Shigeyuki Kondo (Nagoya University) A K3 surface defined over an algebraically closed field is said to be supersingular (in the sense of Shioda) if the rank of its Néron-Severi lattice is 22. Supersingular K3 surfaces exist only when the base field is of positive characteristic. In this talk, we present certain duality between Neron-Severi lattices of supersingular K3 surfaces. ....  ....  On the fixed part of a non-special linear system over a normal surface singularity Kazuhiro Konno (Osaka University) Consider the minimal resolution space X of a normal surface singularity (V,o), and let L be a line bundle with L-K_X nef. Several years ago, I showed that the fixed part of |L| can be contracted to rational singular points. In the talk, I show that any sandwiched singular point arises in this way, and vice versa. ....  ....  Q-Gorenstein deformation and its applications Yognam Lee (Korea Advanced Institute of Science and Technology, Daejon) In this talk we will discuss Q-Gorenstein schemes and Q-Gorenstein morphisms in a general setting. Based on the notion of Q-Gorenstein morphism, we define the notion of Q-Gorenstein deformation and discuss its properties. Versal property of Q-Gorenstein deformation and its applications are also considered. This is joint work with Noboru Nakayama. ....  ....  The orbifold fundamental group of moduli spaces of elliptic surfaces Michael Lönne (Leibniz Universität Hannover) Over the complex numbers we consider different kinds of moduli problems for elliptic surfaces and the associated moduli spaces. Then we show how to solve the problem to get a finite presentation of their orbifold fundamental groups, where braid monodromy is the principal tool. This will be illustrated in the toy case of elliptic curves. If time permits we discuss possible relations with symmetric domains. ....  ....  Autoduality and Fourier-Mukai for compactified Jacobians Margarida Melo (Universidade de Coimbra) Among abelian varieties, Jacobians of smooth projective curves C have the important property of being autodual, i.e., they are canonically isomorphic to their dual abelian varieties. This is equivalent to the existence of a Poincaré line bundle P on J(C)×J(C) which is universal as a family of algebraically trivial line bundles on J(C). A yet other instance of this fact was discovered by S. Mukai, who proved that the Fourier-Mukai transform with kernel P is an auto-equivalence of the bounded derived category of J(C). I will talk on joint work with Filippo Viviani and Antonio Rapagnetta, where we try to generalize both the autoduality result and Mukai´s equivalence result for singular reducible curves X with locally planar singularities. Our results generalize previous work of Arinkin, Esteves, Gagné, Kleiman and can be seen as an instance of the geometric Langlands duality for the Hitchin fibration. ....  ....  Canonical degree of nodal curves on a surface of general type Yoichi Miyaoka (University of Tokyo) Let X be a minimal surface of general type and let C be an irreducible curve of genus g on X. A conjecture of Green-Lang predicts that the canonical degree CKX is bounded by a certain function of (KX)2, c2(X) and g, and it has been verified for curves with bounded number of ordinary nodes and ordinary triple points. We discuss how to address the conjecture for curves with only ordinary nodes. ....  ....  Toward Dirichlet's unit theorem on arithmetic varieties ----- adelic case ----- Atsushi Moriwaki (Kyoto University) In this talk, I would like to discuss a generalization of Dirichlet's unit theorem for adelic arithmetic divisors on arithmetic varieties. ....  ....  Poisson deformations and Mori dream spaces Yoshinori Namikawa (Kyoto University) A normal complex variety is called a symplectic variety if there is a holomorphic symplectic 2-form on its regular locus. In this talk we will consider an affine symplectic variety X having a C*-action with positive weights and its crepant resolution Y. We prove that Y is a relative Mori dream space by using Poisson deformations of Y. ....  ....  Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy Keiji Oguiso (Osaka University) 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. We also give an explanation why we are interested in rational, Calabi-Yau and biregular?. This is a joint work with Doctor Tuyen Trung Truong. ....  ....  Minimal models for Kähler threefolds. Thomas Peternell (Universität Bayreuth) I will report on joint work with Andreas Höring, constructing minimal models for Kähler threefolds whose canonical classes are pseudo-effective. ....  ....  Graded ring constructions of algebraic varieties Miles Reid (University of Warwick) Many of the known constructions of algebraic varieties relate closely to graded ring constructions. This idea includes simple cases such as hypersurfaces or complete intersection in weighted projective spaces, but also more complicated things such as unprojections and structure results for Gorenstein rings in codimension 4. I will discuss some recent progress using these methods. ....  ....  Classification of stable surfaces: first steps Sönke Rollenske (Universität Bielefeld) The Gieseker moduli space of surfaces of general type is not compact but, in analogy to dimension one, it can be embedded into the moduli space of stable surfaces as an open subset (usually not dense). I will present a few motivating examples of such surfaces and explain how we extend some of the classical tools for surfaces of general type to this setting. The talk will be based on joint work with Wenfei Liu and Rita Pardini & Marco Franciosi ....  ....  On (cubic) hypersurfaces with vanishing hessian Francesco Russo (Università di Catania) I will report on recent joint work with Rodrigo Gondim on projective hypersurfaces for which the determinant of the hessian matrix of their equations vanishes identically. These objects have been investigated by P. Gordan and M. Noether in a pioneering fundamental paper where they analyze Hesse's wrong claim according to which cones are the only projective hypersurfaces in $\mathbb P^N$ for which the previous condition holds. The question is quite subtle since Hesse's claim is true for $N\leq 3$, as firstly proved by Gordan and Noether, and in general false for $N\geq 4$. The classification of hypersurfaces with vanishing hessian in $\mathbb P^4$, not cones, was obtained by Gordan-Noether and later independently by A. Franchetta while it remains an open problem for $N\geq 5$. I will survey the known results and try to explain the relevance of hypersurfaces with vanishing hessian not cones for geometry and analysis. Then I will concentrate on the case of cubic hypersurfaces, where we improve the classical approach of U. Perazzo obtaining a Structure Theorem in any dimension and complete classification for $N\leq 6$. If time allows, I will also try to outline the probable classification of hypersurfaces with vanishing hessian of arbitrary degree for $N\leq 6$. ....  ....  K-stability and Kähler-Einstein metric Gang Tian (Beijing University-Princeton University) I will start with the definition of K-stability and discuss how it fits with usual picture in geometric invariant theory. Next I will discuss how it is related to the existence of Kähler-Einstein metrics and some recent progress. ....  ....  The dual complex of singularities and degenerations Chenyang Xu (Beijing University) Using MMP, for isolated singularities and degenerations, we find a canonical representative of the dual complex which is a well defined simplicial space. We will also discuss it connection to the essential skeleton of Berkovich spaces. ....  ....      Organizing and Scientific Committee: Fabrizio Catanese (Universität Bayreuth) Ludmil Katzarkov (Universität Wien) Yujiro Kawamata (University of Tokyo) Roberto Pignatelli (Università di Trento) Gang Tian (Princeton University) Alessandro Verra (Università "Roma Tre") Sponsored by Forschergruppe 790 of the DFG: "Classification of algebraic surfaces and compact complex manifolds" Prin 2010-11: Geometria delle varietà algebriche GNSAGA Gruppo Nazionale INDAM Futuro in Ricerca 2012: Moduli Spaces and Applications