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Hyperkähler manifolds are higher-dimensional analogs of K3 surfaces. Verbitsky and Markmann recently proved that their period map is an open embedding. In a joint work with E. Macri, we explicitly determine the image of this map in some cases. I will explain this result together with a nice application (found by Bayer and Mongardi) to the (almost complete) determination of the image of the period map for cubic fourfolds, hereby partially recovering a result of Laza.
Hyperkähler manifolds are higher-dimensional analogs of K3 surfaces. Verbitsky and Markmann recently proved that their period map is an open embedding. In a joint work with E. Macri, we explicitly determine the image of this map in some cases. I will explain this result together with a nice application (found by Bayer and Mongardi) to the (almost complete) determination of the image of the period map for cubic fourfolds, hereby partially ...

14C34 ; 14E07 ; 14J50 ; 14J60

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Twistor spaces of K3 surfaces are non-Kähler compact complex manifolds which play a fundamental role in the moduli theory of K3 surfaces. They come equipped with a holomorphic submersion to the complex projective line which under the period map corresponds to a twistor line in the K3-period domain. In this talk I will explain how one can view a twistor line as a certain base point in the linear cycle space of the period domain. Then, based on joint work in progress with Daniel Greb, Tim Kirschner and Martin Schwald I will present new results concerning the deformations of twistor spaces of K3 surfaces and their relation to the cycle space of the period domain.
Twistor spaces of K3 surfaces are non-Kähler compact complex manifolds which play a fundamental role in the moduli theory of K3 surfaces. They come equipped with a holomorphic submersion to the complex projective line which under the period map corresponds to a twistor line in the K3-period domain. In this talk I will explain how one can view a twistor line as a certain base point in the linear cycle space of the period domain. Then, based on ...

14J28 ; 14J60 ; 14C25 ; 53C26 ; 53C28

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A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet bundles. Shortly afterwards, Ya Deng obtained effective degree bounds by means of a refined technique. Our goal here will be to explain a drastically simpler proof that yields an improved (though still non optimal) degree bound, e.g. $d_n=[(en)^{2n+2}/5]$. We will also present a more general approach that could possibly lead to optimal bounds.
A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet ...

32Q45 ; 32L10 ; 53C55 ; 14J40

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Simpson’s classic nonabelian Hodge correspondence establishes an equivalence of categories between local systems on a projective manifold, and certain Higgs sheaves on that manifold. This talk surveys recent generalisations of Simpson’s correspondence to the context of projective varieties with klt singularities. Perhaps somewhat surprisingly, these spaces exhibit two correspondences: one pertaining to local systems on the whole space, and one to local systems on its smooth locus. As one application, we resolve the quasi-étale uniformisation problem for minimal varieties of general type, and to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, co-compact groups of automorphisms that act freely in codimension one.
Simpson’s classic nonabelian Hodge correspondence establishes an equivalence of categories between local systems on a projective manifold, and certain Higgs sheaves on that manifold. This talk surveys recent generalisations of Simpson’s correspondence to the context of projective varieties with klt singularities. Perhaps somewhat surprisingly, these spaces exhibit two correspondences: one pertaining to local systems on the whole space, and one ...

14E30 ; 53C07 ; 32Q30 ; 14E20 ; 32Q26

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The plane Cremona group is the group of birational transformations of the projective plane. I would like to discuss why over algebraically closed fields there are no homomorphisms from the plane Cremona group to a finite group, but for certain non-closed fields there are (in fact there are many). This is joint work with Stéphane Lamy.

14E07 ; 14E30

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The gonality of a variety is defined as the minimal gonality of curve sitting in the variety. We prove that the gonality of a very general abelian variety of dimension $g$ goes to infinity with $g$. We use for this a (straightforward) generalization of a method due to Pirola that we will describe. The method also leads to a number of other applications concerning $0$-cycles modulo rational equivalence on very general abelian varieties.

14C15 ; 14C25 ; 14J70 ; 14J28 ; 14H51 ; 14Kxx

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