Exposés

Résumé :
  Jet bundles and reparametrization group actions occupy a central role in hyperbolicity questions due to the strategy developed by Demailly, Siu, et al., to study the degeneracy of entire curves in projective varieties. We explain how this strategy ultimately reduces the problem to the intersection theory of non-reductive geometric invariant theory-type quotients, which has led to the recent proof of the Green-Griffiths-Lang and Kobayashi hyperbolicity conjectures for generic projective hypersurfaces with effective degree bounds.

Résumé :
  Consider a homogeneous polynomial \(P\) with integer coefficents. Its “naive” height is, in classical Diophantine geometry, defined as the maximum of the absolute values of the coefficients of \(P\). More generally, in the framework of Arakelov geometry, the height is a real number, depending on the choice of a Hermitian metric on the hyperplane line bundle over the complex projective variety \(X\), cut out by \(P\). In a recent joint work with Rolf Andreasson we introduce a canonical height, obtained by taking the metric in question to be Kähler-Einstein - if such a metric exists. This canonical height has several useful properties. In particular, it can be expressed as a limit of periods on the \(N\)-fold products \(X^{N}\), as \(N\) tends to infinity. In this talk I will explain how this leads to an explicit expression for the canonical height of any diagonal homogeneous polynomial \(P\) in three variables. The formula involves the Hurwitz zeta function and its derivative at \(s=-1\). Some connections to Shimura curves and Parshin's conjectural arithmetic Miyaoka–Yau type inequality will also be pointed out. No prior knowledge of Arakelov geometry will be assumed.

Résumé :
  Some 20 years ago, Arezzo and Pacard established that the existence of extremal Kähler metrics on a compact Kähler manifold is preserved under certain point blowups. This was later extended by Seyyadili-Szekelyhidi to blowups of certain higher dimensional submanifolds, and, quite recently, to the case of weighted extremal metrics, by Hallam. In this talk we present a new approach unifying both cases, based on the properness of the Mabuchi K-energy. This is joint work with Mattias Jonsson and Antonio Trusiani.

Résumé :
  Ramified Galois covers with finite abelian groups are frequently used in complex algebraic geometry, for example in the proof of vanishing theorems. They behave differently in positive characteristics, as there exists no equivariant resolution of singularities in this setting. The talk will propose a remedy to that failure, and describe a class of diagonalizable covers with applications to curves and surfaces.

Résumé :
  Let \(X\) be a smooth, complex Fano 4-fold, and \(\rho(X)\) its Picard number. We will discuss the following theorem: if \(\rho(X)>12\), then \(X\) is a product of del Pezzo surfaces. This implies, in particular, that the maximal Picard number of a Fano 4-fold is 18. After an introduction and a discussion of examples, we explain some of the ideas and techniques involved in the proof.

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  Envelopes are key objects in pluripotential theory. In this talk I will introduce envelopes on a compact Kähler manifold and I will study their regularity. In particular, I will show that, even in singular contexts, they are \(C^{1,\alpha}\) on a Zariski open subset. This is based on a joint work with S. Trapani.

Résumé :
  In his article published in the Gazette des Mathématiciens (1991), Jean-Pierre Demailly asked a question about the definition of the intersection of positive closed currents of higher bi-degrees. In this talk, I will discuss some answers to this problem as well as some applications in complex dynamics.

Résumé :
  In his wonderful book "Shafarevich maps and automorphic forms", J. Kollár asked almost 30 years ago for a "good" birational version of the notion of Kähler hyperbolicity introduced by M. Gromov in the early '90s. Kähler hyperbolic manifolds are compact Kähler manifold admitting a Kähler form whose pull-back to the universal cover becomes d-exact and moreover with a bounded primitive. Such manifolds are of general type (more than this: with ample canonical bundle), Kobayashi hyperbolic, and with large fundamental group. We shall report on the recently intruduced class of weakly Kähler hyperbolic manifolds which indeed provides such a birational generalization. Among other things, we shall explain that these manifolds are of general type, satisfy a precise quantitative version of the Green-Griffiths conjecture and have generically arbitrarily large fundamental group. Moreover, their properties permit to verify Lang's conjecture for Kähler hyperbolic manifolds. This is a joint work with F. Bei, B. Claudon, P. Eyssidieux, and S. Trapani.

Résumé :
  In this recent joint work, we prove that a stable, reflexive \(\mathbb Q\)-sheaf \(\mathcal F\) on a compact Kähler threefold with log terminal singularities, the orbifold Chern classes of \(\mathcal F\) satisfy the Bogomolov-Gieseker inequality. The proof involves two main steps. First, one constructs and studies global, singular Kähler metrics which are orbifold Kähler metrics on the orbifold locus. Then, one shows that the Chern-Weil currents associated to the singular Hermite-Einstein metric on \(\mathcal F\) are \(\bar\partial\) closed on the orbifold locus.

Résumé :
  A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In a joint work with Saverio Secci we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, hence has codimension two, all the anticanonical divisors are singular. In this talk I will explain how this statement is related to extension problems on \(K\)-trivial varieties with a fibre space structure.

Résumé :
  We say that the formal principle with convergence holds for a compact complex submanifold \(A\) in a complex manifold \(X\), if any formal isomorphism between the formal neighborhood of \(A\) in \(X\) and that of another complex submanifold \(A'\) in another complex manifold \(X'\) always converges, namely, the formal isomorphism can be extended to a biholomorphic map between suitable neighborhoods. Hirschowitz and Commichau-Grauert showed that the formal principle with convergence holds if the normal bundle of the submanifold is sufficiently positive. We discuss the problem when the normal bundle is only weakly positive, but the submanifold satisfies certain geometric conditions. Our main interest is when the submanifold is a general minimal rational curve in a uniruled projective manifold, such as a general line on a rational homogeneous space or a projective hypersurface of low degree.

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  I will survey a body of work on invariants measuring how far a given variety is from being rational, and how far two varieties are from being birationally isomorphic

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  I will discuss the concept of canonical extensions on a projective or compact Kähler manifold and explain the connection to manifolds \(X\) with nef tangent bundles, in particular, when \(X\) is Fano.

Résumé :
  Campana introduced the class of special varieties which conjecturally correspond to varieties antithetic to hyperbolic varieties. We will present new evidences for this conjecture based on recent works with Bartsch-Javanpeykar and with Kebekus.

Résumé :
  I will explain a Hodge-theoretic proof for Hwang's theorem, which says that if the base of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is smooth, then it must be projective space. The result is contained in a joint paper with Ben Bakker from last fall.

Résumé :
  The volume of a line bundle on a smooth projective variety is a rough measure for the asymptotic growth of the dimension of the space of sections of its high tensor powers, and line bundles with positive volume are called big. The volume extends naturally to a continuous function on the real Neron-Severi group, which vanishes outside the big cone and is \(C^1\) differentiable inside of it, by work of Boucksom-Favre-Jonsson and Lazarsfeld-Mustata. An interesting question is then to determine what the optimal regularity is, up to the boundary of the big cone. I will discuss joint work with Simion Filip and John Lesieutre where we construct examples where this volume function is \(C^1\) but not better (at the boundary), and use this to answer negatively a number of questions by Lazarsfeld and others.

Résumé :
  Numerical Bogomolov sheaves are rank one coherent subsheaves \(L\) of \(\Omega_X^p\) having numerical dimension \(p\) for some \(p\) > 0, and a complex projective (or more generally compact Kähler) manifold \(X\). The existence of \(L\) like above determines a distribution \(\mathcal D\), namely the kernel of \(L\). According to a theorem of Jean-Pierre Demailly, this distribution is actually integrable. The main part of this talk will be devoted to questions (essentially unsolved) concerning the structure of this foliation and are motivated on one hand by some results concerning the case \(p=1\), on the other hand by a recent joint work with Benoît Claudon.

Résumé :
  One major new birational geometry problem arising in understanding stable degeneration of varieties, which is the algebraic analogue to the compactness of Kähler-Einstein type metrics, is finite generation for valuations of higher rational rank. In the past a few years, we have established finite generation for minimizing valuations of various functionals, by first showing those minimizers are ‘special’; and then proving any special valuation satisfies finite generation. This raises the question on how ‘special’ valuations are distributed, which still remain to be widely open. In this talk, I will report results and questions along this direction. (Based on joint work with Yuchen Liu and Ziquan Zhuang).