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Title and Abstract
1. Speaker : Sebastien Boucksom (Paris)
Title : "An introduction to Okounkov bodies"
Abstract :
The theory of Okounkov bodies associates to any line bundle over a projective algebraic variety a convex body, in such a way that couting the number of lattice points in the scaled body computes the asymptotics of the number of sections of large powers of the line bundle. This technique, which generalizes the usual picture from toric geometry, enables to recover in a elementary way (and independently of the characteristic of the base field) several fundamental results about big line bundles, as well as their extension to arbitrary line bundles and more general graded linear series, which had been previously out of reach. I will present in detail the heart of the theory, which is remarkably simple, along with some applications to algebraic and arithmetic geometry.
References :
1. R. LAZARSFELD, M. MUSTATA – Convex bodies associated to linear series. Ann. Sci. Ec. Norm. Super. (4) 42 (2009), 783–835.
2. K. KAVEH, A. KHOVANSKII – Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Preprint (2012) arXiv :0904.3350v3. To appear in Ann. Math.
3. S. BOUCKSOM - Corps d'Okounkov. Notes of a talk at the Séminaire Bourbaki (in French!), available at http://www.math.jussieu.fr/~boucksom/publis.html
2. Speaker : Vincent Guedj (Toulouse)
Title : "Geometry and Topology of the space of Kaehler metrics"
Abstract :
"Let X be a compact Kaehler manifold. The space of Kaehler metrics in a fixed cohomology class can be formally regarded as a non positively curved infinite dimensional locally symmetric space. We shall describe the corresponding Mabuchi L2 metric, geodesics and try and understand its metric completion".
References :
1. Mabuchi, Osaka J Math, 1987
2. Semmes, AM J Math, 1992
3. Donaldson, AMS, 1999
4. Chen, JDG, 2000
5. Phong-Sturm, Inventiones, 2006
6. Guedj ed. Lecture Notes Math, vol 2038, 2012
7. Lempert-Vivas, Duke Math J., 2013
3. Speaker : Mattias Jonsson (Michigan University)
Title : An introduction to Berkovich spaces
Abstract :
A complex algebraic varieties is defined by the zero locus of polynomials with complex coefficients. It can be viewed as a complex manifold (or analytic se) and can be studied using analytical tools.
When studing polynomials with coefficients in a general normed field (such as the p-adic numbers), the role of complex manifolds or analytic sets is played by Berkovich spaces.
I will give an introduction to Berkovich spaces in general and explain how concepts such as metrics of line bundles behave in this context.
References :
1. M. Baker, and R. Rumely. Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs, 159. American Mathematical Society, Providence, RI, 2010.
2. V. Berkovich. Spectral theory and analytic geometry over non-Archimedean _elds.Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990.
3. S. Boucksom, C. Favre and M. Jonsson. Singular semipositive metrics in non-Archimedean geometry. arXiv.org:1201.0187.
4. S. Boucksom, C. Favre and M. Jonsson. Solution to a non-Archimedean Monge-Amp_ere equation. arXiv.org:1201.0188.
4. Speaker : Ahmed Zeriahi (Toulouse)
Title : Stability of solutions to degenerate complex Monge-Ampère equations
Abstract :
We will consider degenerate complex Monge-Ampère equations arising in the problem of finding singular Kähler-Einstein metrics on compact Kähler manifolds with mild singularities.
In the smooth and non degenerate case, these equations was solved in late seventies by S.T. Yau in the case when the first Chern class of the compact manifold is zero in connection with the celebrate Calabi's conjecture and by T. Aubin and S.T. Yau ( independently ) when the first Chern class of the compact manifold is negative.
The main step in their proof is an a priori $C^0$-estimate of the solution of the corresponding equation. In late nineties S. Koldziej gave a new "pluripotential proof" of Yau's a priori $C^0$-estimate and corresponding stability results which allow him to solve more general equations.
We will explain how pluripotential theory provides a priori estimates and stability results for weak solutions of degenerate complex Monge-Ampère equations in a quite general setting. This provides a new method for proving existence of weak solutions to these equations without using the continuity method.
References :
1. Kołodziej, Sławomir : The Monge-Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52 (2003), no. 3, 667–686.
2. Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed : Singular Kähler-Einstein metrics. J. Amer. Math. Soc. 22 (2009), no. 3, 607–639.