Schedule
AbstractsGuedj & Guenancia: Families of Kähler-Einstein metricsIn this lecture, we will develop analytic techniques to study proper, holomorphic surjective maps f:X\to Y between Kähler manifolds. A main case of interest will be when general fibers X_y admit a possibly singular Kähler-Einstein metric, for instance if X_y is of general type. In that case, we will explore the properties of the so-called relative Kähler-Einstein metric \omega_{X/Y} by showing that it is a positive current on X with minimal singularities in some sense. Lamboglia & Mazzon: Perspectives on tropical and non-Archimedean geometryTropical and non-Archimedean geometry is a novel and fast-moving research area in mathematics. Its techniques provide new approaches to investigate classical problems in algebraic geometry. In this talk we aim at presenting some key ideas in these fields and their applications. Along with a short introduction to both topics, we will highlight evidences of their deep connection. We will then show how techniques from tropical and non-Archimedean geometry give a new insight in the study of Fano schemes of varieties, and of degenerations of hyper-Kähler manifolds. Bernasconi & Patakfalvi: Positive characteristic algebraic geometryIn this lecture we will survey modern developments in algebraic geometry over fields of positive characteristic, with an emphasis on the classification theory. After highlighting special features of characteristic p, we will discuss recent results on failure of Kodaira vanishing and its connection to the minimal model program, the geometry of Mori fibre spaces (especially in dimension three) and varieties with trivial canonical class. Perrin & Zucconi: On the geometry of the quintic del Pezzo threefold[I part:] In the first part of the talk we will review some aspects of the geometry of the conics of the quintic del Pezzo 3-fold. In particular we will present a generalisation of the Mukai’s description of the genus 12 Fano $3$-fold. [II part:] In the second part of the talk we will give a definition of rational simple connectedness and explain how to use lines and conics to prove it for the quintic del Pezzo 3-fold. O’Grady & Voisin: (Hyper-)Kähler manifoldsThere are very few projective complex manifolds that deform to compact Kähler manifolds with no topologically nontrivial holomorphic line bundles. Complex tori are an example, and also hyper-Kähler manifolds. In the first part of the talk we will discuss conversely a criterion for a compact Kähler manifold to have projective deformations. In the second part, after a very brief introduction to hyper-Kähler manifolds, we will present a recent result on moduli of vector bundles on projective hyper-Kählers. Castravet & Martinelli: Gale duality, blowups and moduli spacesThe Gale correspondence provides a duality between sets of $n$ points in projective spaces $\mathbb{P}^s$ and $\mathbb{P}^r$ when $n=r+s+2$. For small values of $s$, this duality has a remarkable geometric manifestation: the blowup of $\mathbb{P}^r$ at $n$ points can be realized as a moduli space of vector bundles on the blowup of $\mathbb{P}^s$ at the Gale dual points. We explore this realization to describe the birational geometry of blowups of projective spaces at points in very general position. We will survey known results in this area in connection with moduli problems and in the second part we will focus on the cases where the blowup fails to be a Mori Dream Space, reporting on a joint work with Carolina Araujo and Inder Kaur. Pasquinelli & Riolo: Real and complex hyperbolic cone-manifoldsThurston's hyperbolic Dehn filling was a striking result in (real) 3-dimensional topology, showing that many 3-manifolds are hyperbolic. The proof involves special deformations of hyperbolic "cone-manifolds", i.e. hyperbolic metrics with cone-like singularities, giving rise to some topological surgeries. In another work, Thurston showed that the complex hyperbolic n-orbifolds associated to the Deligne-Mostow lattices in PU(n,1) arise as special points of a continuous family of complex hyperbolic cone-manifolds. Any two orbifolds of the family are topologically related by surgery. We will explain this phenomenon in a unified way and give an idea of what is known, with special emphasis in real dimension four and complex dimension two. Arosio & Lo Bianco: Dynamics on complex manifolds: rigidity and flexibilityIn this talk we will discuss some properties of automorphisms of complex manifolds from the point of view of dynamical systems. In particular, we will highlight some rigidity phenomena in the projective case as opposed to the flexibility in the non-compact case. In the projective case, we will see that the action of the automorphism in cohomology has a striking link with its dynamics; such link is particularly clear in small dimensions. We will also see some instances of rigidity: existence of preserved fibrations, finiteness of automorphisms for varieties of general type, algebraic structure of (the connected component of) the group of automorphisms, bounds on its size. In the non-compact case, we will focus on the dynamics on manifolds like C^n, which has an infinite-dimensional family of automorphisms, or more generally manifolds with the density property, where Andersén-Lempert’s theory is available. This has strong consequences on the dynamics: for example we will see that a generic volume preserving automorphism of C^n is chaotic, and that one obtains cohomological restrictions when trying to generalize this result to complex manifolds with the density property. | |||||||||||||||||||||||||||||||||||
Geometry in Pairs
19th and 20th of December 2019, Roma Tre
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