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# Documents  Floris, Enrica | enregistrements trouvés : 7

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## Failure of the Brauer-Manin obstruction for a simply connected fourfold, and an orbifold version of the Mordell theorem Kebekus, Stefan | CIRM H

Post-edited

Research talks;Algebraic and Complex Geometry

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov’s etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.
This talk discusses a construction of simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana’s "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
Joint work with Pereira and Smeets.
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov’s etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.
This talk discusses a construction of simply connected fourfolds over global fields of positive characteristic for ...

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## Invariance of plurigenera for foliations on surfaces Floris, Enrica | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

Let $X$ be a smooth algebraic surface. A foliation $F$ on $X$ is, roughly speaking, a subline bundle $T_F$ of the tangent bundle of $X$. The dual of $T_F$ is called the canonical bundle of the foliation $K_F$. In the last few years birational methods have been successfully used in order to study foliations. More precisely, geometric properties of the foliation are translated into properties of the canonical bundle of the foliation. One of the most important invariants describing the properties of a line bundle $L$ is its Kodaira dimension $\kappa(L)$, which measures the growth of the global sections of $L$ and its tensor powers. The Kodaira dimension of a foliation $F$ is defined as the Kodaira dimension of its canonical bundle $\kappa(K_F)$. In their fundamental works, Brunella and McQuillan give a classfication of foliations on surfaces on the model of Enriques-Kodaira classification of surfaces. The next step is the study of the behaviour of families of foliations. Brunella proves that, for a family of foliations $(X_t, F_t)$ of dimension one on surfaces, satisfying certain hypotheses of regularity, the Kodaira dimension of the foliation does not depend on $t$. By analogy with Siu's Invariance of Plurigenera, it is natural to ask whether for a family of foliations $(X_t, F_t)$ the dimensions of global sections of the canonical bundle and its powers depend on $t$. In this talk we will discuss to which extent an Invariance of Plurigenera for foliations is true and under which hypotheses on the family of foliations it holds. Let $X$ be a smooth algebraic surface. A foliation $F$ on $X$ is, roughly speaking, a subline bundle $T_F$ of the tangent bundle of $X$. The dual of $T_F$ is called the canonical bundle of the foliation $K_F$. In the last few years birational methods have been successfully used in order to study foliations. More precisely, geometric properties of the foliation are translated into properties of the canonical bundle of the foliation. One of the ...

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## On the B-Semiampleness Conjecture Floris, Enrica | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

An lc-trivial fibration $f : (X, B) \to Y$ 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 informations on the singular fibres; a divisor, called moduli part, that contains informations 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 dimension. This is a joint work with Vladimir Lazić. An lc-trivial fibration $f : (X, B) \to Y$ 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 informations on the singular fibres; a divisor, called moduli ...

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## Momentum polytopes of Kähler compact multiplicity-free manifolds Cupit-Foutou, Stéphanie | CIRM H

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Research talks;Algebraic and Complex Geometry;Lie Theory and Generalizations

I will present some results about the momentum polytopes of the multiplicity-free Hamiltonian compact manifolds acted on by a compact group which are Kählerizable. I shall give a characterization of these polytopes, explain how much they determine these manifolds and sketch some applications of this characterization - most of these results have been obtained jointly with G. Pezzini and B. Van Steirteghem.

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## Rational curves on K3 surfaces Gounelas, Frank | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some of the main previously known cases more conceptually and extending the result to arbitrary genus in a suitable sense. Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some ...

14J28

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## Representation varieties of fundamental groups of complex algebraic varieties and mixed Hodge structures Lefèvre, Louis-Clément | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry;Topology

We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible fundamental groups of complex algebraic varieties. This was first described by Goldman and Millson in the case of compact Kähler manifold, using formal deformation theory and differential graded Lie algebras. We review this using methods of Hodge theory and of derived deformation theory and we are able to describe locally the representation variety for non-compact smooth varieties and representations underlying a variation of Hodge structure. We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible ...

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## The essential skeletons of pairs and the geometric P=W conjecture Mauri, Mirko | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new theory of essential skeletons over a trivially-valued field. As a byproduct, inspired by these constructions, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds. In this talk we will explain how these new non-archimedean techniques can shed new light into classical algebraic geometry problems. The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new ...

14G22

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