Random holomorphic sections associated with asequence of line bundles on compact kähler manıfolds

The study of zeros of random polynomials is a fascinating subject due to its numerousconnections within mathematics and physics. In particular, the distribution of thesezeros is crucial for understanding chaotic dynamics and quantum ergodicity, as itmodels the behavior of nodal sets of eigenfunctions in chaotic quantum systems.Building upon these ideas, the concepts naturally extend to higher dimensionsthrough random holomorphic sections, which generalize random polynomials, givingrise to the emerging field of stochastic Kähler geometry. This thesis investigates twointerconnected problems within the realm of stochastic Kähler geometry, focusingon the equidistribution and statistical fluctuations of zeros of random holomorphicsections associated with Hermitian holomorphic line bundles on compact Kählermanifolds.In the first part, we establish an equidistribution phenomenon for zeros of systemsof random holomorphic sections associated with a sequence of positive Hermitianholomorphic line bundles with C 2 metrics on a compact Kähler manifold X. This isachieved through variance estimates and an analysis of the expected distributions ofrandom zero currents of integration in any codimension k. Our results extend previousfindings in the field by encompassing a broader range of probability distributions,including Gaussian, Fubini-Study measures, and probability measures with boundeddensities and logarithmically decaying tails. In the second part, we establish a central limit theorem for random currents ofintegration along the zero divisors of standard Gaussian holomorphic sections. Thistheorem, proved within the framework of sequences of holomorphic line bundles,demonstrates the asymptotic normality of smooth linear statistics of random zerodivisors. Along the way, using methods from complex differential geometry, such asDemailly’s L2-estimates for the ¯∂-operator, we obtain first-order asymptotics andupper decaying estimates for near and off-diagonal Bergman kernels.