Abstracts
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Olivier Benoist: Curves on real rationally connected varieties
The set of real points X(R) of a smooth projective real algebraic variety X is a compact manifold. Is it possible to approximate loops in X(R) by real loci of algebraic curves in X? This property fails in general, but might hold if X is rationally connected. In this talk, I will provide positive answers in particular cases that include cubic hypersurfaces and compactifications of homogeneous spaces under connected linear algebraic groups. This is joint work with Olivier Wittenberg.
Cinzia Casagrande: Fano 4-folds with rational fibrations
Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. In the talk we will focus on Fano 4-folds with large second Betti number b2, studied via birational geometry and the detailed study of their contractions and rational contractions. We recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a (regular) contraction.
The main result that we want to present is the following: let X be a Fano 4-fold having a rational contraction X→Y of fiber type (with dim Y > 0). Then either X is a product of surfaces, or b2(X) is at most 18, or Y is P1 or P2.
Benoît Claudon: Fundamental groups of compact Kähler threefolds
This talk will be concerned with the Kodaira problem for the fundamental group which consists in asking whether the fundamental group of a compact Kähler manifold can be also realized as the fundamental group of a smooth projective variety. I will explain how to get a positive answer to this question in dimension 3 (joint work with Hsueh-Yung Lin and Andreas Höring).
Jean-Pierre Demailly: Bergman bundles and transcendental holomorphic Morse inequalities
The talk will present a few analytic techniques that are useful to investigate the geometry of non necessarily projective compact complex manifolds. One of this tool consists of Bergman bundles: they are holomorphic Hilbert bundles that turn out to be always very ample in some sense, even though they can be made locally trivial only as real analytic vector bundles. We will also discuss some results in the direction of the (still conjectural) transcendental holomorphic Morse inequalities.
Simone Diverio: About Lang’s conjecture for quotients of bounded domains
We shall discuss how to verify Lang’s conjecture (a projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties) for the following class of manifolds: projective manifolds which admit an étale cover with a bounded strictly plurisubharmonic function on it. In particular, compact free quotients of bounded domains are in this class. This is a joint work with S. Boucksom.
Stéphane Druel: Characterization of generic projective space bundles and application to foliations
The existence of sufficiently positive subsheaves of the tangent bundle of a complex projective manifold imposes strong restrictions on the manifold. In particular, several special varieties can be characterized by positivity properties of their tangent bundle. In this talk we consider distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. This is joint work with Carolina Araujo.
Enrica Floris: On the b-Semiampleness conjecture
An lc-trivial fibration f: (X,B)→Y is a fibration such that the log-canonical divisor of the pair (X,B) is trivial along the fibres of f. As in the case of the canonical bundle formula for elliptic fibrations, the log canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of Y; a divisor, called discriminant, which contains information on the singular fibres; a divisor, called moduli part, that contains information on the variation in moduli of the fibres. The moduli part is conjectured to be semiample. Ambro proved the conjecture when the base Y is a curve. In this talk we will explain how to prove that the restriction of the moduli part to a hypersurface is semiample assuming the conjecture in lower dimensions. This is a joint work with Vladimir Lazić.
Aleksei Golota: On positivity of the conormal bundles to foliations on threefolds
Consider a codimension one singular foliation on a complex projective threefold. In my talk I will discuss certain connections between geometry of the threefold and curvature properties of general leaves of the foliation. The main guiding principle is to use positivity of the conormal bundle to the foliation.
Daniel Greb: Canonical complex extensions of compact Kähler manifolds
Given a (compact) complex manifold X, any Kähler class defines an affine bundle over X, and any Kähler form in the given class defines a totally real embedding of X into this affine bundle. I will formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of X. For compact Kähler manifolds of non-negative holomorphic bisectional curvature, I will explain the close relation of this construction to adapted complex structures in the sense of Lempert-Szőke and to the existence question for good complexifications in the sense of Totaro. Moreover, I will study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. This is joint work with Michael Wong.
Henri Guenancia: Quasi-projective manifolds with negative holomorphic sectional curvature
I will explain the following result. Let X be a smooth projective manifold and let D be a divisor with simple normal crossings. If X\D admits a Kähler metric with holomorphic sectional curvature bounded above by a negative constant, then KX+D is big. Using that result, one can verify a conjecture of Lang relating hyperbolicity of a variety and the geometry of its subvarieties in the particular case of compact Kähler manifolds with negative holomorphic sectional curvature.
Andreas Höring: Pseudoeffective thresholds for polarised Hyperkähler manifolds
Let X be a complex projective manifold that is Hyperkähler. By a recent result of Thomas Peternell and myself the cotangent bundle of X is not pseudoeffective. One way to mesure this negativity more precisely is to give sufficient conditions for the twist of ΩX ⊗ A to be pseudoeffective if A is some ample line bundle. I will report on work in progress with Fabrizio Anella: for Hyperkähler manifolds of type K3[n], we give a sufficient condition that depends only on the Segre classes/Beauville form of X. For K3 surfaces this sufficient condition is necessary for infinitely many 19-dimensional families.
Vladimir Lazić: Basepoint freeness beyond the Basepoint free theorem
I will discuss nonvanishing and semiampleness conjectures and results which go beyond the Abundance Conjecture and the theorems of Shokurov and Kawamata. This is joint work with Thomas Peternell.
Christian Lehn: Kawamata-Morrison cone conjecture for singular symplectic varieties
In a joint work in progress with G. Mongardi and G. Pacienza, we study the Kawamata-Morrison cone conjecture and its birational analog for singular symplectic varieties. As an application of the deformation theoretic results developed with Pacienza and the Global Torelli theorem for singular varieties proven with B. Bakker, we get insight into the structure of the nef and movable cones of a singular symplectic variety.
Zsolt Patakfalvi: Semi-positivity of the anti-relative canonical divisor for families of uniformly K-stable Fanos
The relative canonical divisor of a family of canonically polarized varieties (with mild singularities) is semi-positive by results that are considered "classical by now". However, if one switches all signs, there is not clear behavior in general. That is, the relative anti-canonical divisor of a family of Fano varieties (with mild singularities) is almost never semi-positive, except a few degenerate cases. I will present results from my joint work with Giulio Codogni about how one can still show semi-positivity in the latter case by requiring the general fibers to by uniformly K-stable and by involving a multiple of the CM line bundle from the base. The key here is that this multiple depends only on invariants of very general fibers. I will give applications too, for example, deducing positivity of the CM line bundle from its semi-positivity, which is an important conjecture in the moduli theory of Fanos.
Thomas Peternell: The algebraic dimension of a potential complex 6-sphere
I will present a proof that a potential complex 6-sphere does not carry any non-constant meromorphic function (joint work with F. Campana and J.-P. Demailly).
Alessandra Sarti: On non-symplectic automorphisms of K3 surfaces
Automorphisms of K3 surfaces were very much studied in the last years. Depending on the action on the holomorphic two form which can be trivial or not, they are called symplectic or non-symplectic. The aim of the talk is to present recent results in the study of non-symplectic automorphisms of 2-power order. In particular in the case of the order 16 I completely describe the families of K3 surfaces carrying such automorphisms.
Stefan Schreieder: On deformations of quintic and septic hypersurfaces
An old question of Mori asks whether in dimension at least three, any smooth specialization of a hypersurface of prime degree is again a hypersurface. A positive answer to this question has up till now only been known in degrees two and three. In this talk I explain how to settle the case of quintics (in arbitrary dimension) and septics in dimension three. Our results follow from numerical characterizations of the corresponding hypersurfaces. In the case of quintics, this extends famous work of Horikawa who analysed deformations of quintic surfaces. This is joint work with J.C. Ottem.
Luca Tasin: Shokurov's polytopes of the moduli space of curves
Let M be the moduli space of stable curves of genus g. The aim of this talk is to explain a partial Shokurov's decomposition of the space of effective boundary divisors D on M describing the log canonical models of K+D. To this aim we introduce many new modular compactifications of the moduli space of smooth curves which are related to the minimal model program of M. This is a joint work in progress with Giulio Codogni and Filippo Viviani.
Nikolaos Tsakanikas: On generalised polarised pairs and termination of flips
In this talk I will present a result concerning the termination of flips. It builds on recent progress on generalised polarised pairs and the relation between the existence of minimal models and weak Zariski decompositions. This is joint work in progress with Vladimir Lazić.
Susanna Zimmermann: Non-simplicity of Cremona groups
The Cremona group is the group of birational transformations of the projective space. Being the symmetry group of the nicest projective variety, it is a classical object in algebraic geometry, and has been studied on and off for more than a century. Already in 1895 Enriques posed the question, as to whether the Cremona groups are simple (contains strict proper normal subgroups). For the plane Cremona group, the question was answered in 2013 by S. Cantat and S. Lamy. I will present that the Cremona groups in higher dimension are not simple as well. This is joint work with J. Blanc and S. Lamy.