Distribution of zeros of random holomorphic sections

报告人：George Marinescu

时间：4月9日（星期三）13:00-14:30

地点：数学楼102报告厅

报告人简介：

Professor George Marinescu is a professor at the University of Cologne. He received his Ph.D. degree from Paris Diderot University (University of Paris 7), and worked as a postdoc at the University of Edinburgh, the Institut de mathématiques de Jussieu, and the Humboldt University of Berlin. Before becoming a professor at the University of Cologne in 2006, he was an assistant researcher at the Humboldt University from 2000 to 2005.

Professor Marinescu is a world renown expert in complex geometry, global analysis, and spectral theory. He has published over 70 paper in journals such as Ann. Sci. Éc. Norm. Supér., JDG, Crelle, CMP, JFA, Adv. Math., JMPA, Math. Ann.. He was awarded, jointly with Xiaonan Ma, the Ferran Sunyer i Balaguer Prize in 2006 for their book "Holomorphic Morse inequalities and Bergman kernels", and he also received the Romanian Academy's 2012 Simion Stoilow Prize.

报告题目：

Distribution of zeros of random holomorphic sections

报告简介：

Kac random polynomials in the complex plane are polynomials of degree whose coefficients are independent, identically distributed random variables. If the coefficients are complex Gaussians with mean zero and variance one, then a classical result of Hammersley shows that the zeros of such random polynomials tend to concentrate on the unit circle as the degree approaches infinity.

We begin with a precise statement of this result, as well as a proof that not only explains the role of the unit circle, but also serves as a model for far-reaching generalizations. A conceptual and very fruitful approach, which can be seen in this proof, is to introduce an inner product on the space of polynomials, which induces the Gaussian probability measure on the space of coefficients, and to consider the asymptotics of the Bergman kernel associated with this inner product.

Polynomials of degree at most generalize to holomorphic sections of the -th power of an ample line bundle →
 over a complex projective manifold

of dimension n. If the bundle has a Hermitian metric h whose Chern curvature

is a Kähler form, then the zeros of the random holomorphic sections converge to in the sense of currents, as . If the curvature is not necessarily positive, then the limit is a certain equilibrium (1,1)-form on . This result actually holds for singular Hermitian metrics on big line bundles, and one can estimate the speed of convergence using a method of Dinh-Sibony. The tools we will use are the following: basic notions of elementary probability theory, elements of complex geometry and pluripotential theory (currents,L² -estimates for ), and in particular the asymptotics of the Bergman kernel function.

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