Distribution of random degeneracy sets for Grassmannian embeddings

Speakers: Dan Coman (Syracuse University)\n\nLet (X,ω) be a compact Kähler manifold, (L,hL) be a positive line bundle, and (E,hE) be a Hermitian holomorphic vector bundle of rank r on X. We show that the pullback by the Kodaira embedding associated to Lp⊗E of the k-th Chern form of the dual universal bundle over the Grassmannian converges as p→∞ to the k-th power of the Chern form c1​(L,hL), for 0≤k≤r. The degeneracy set of a k-tuple of holomorphic sections of Lp⊗E is the locus of points in X where they are linearly dependent. We compute the expectation of the currents of integration along degeneracy sets of random k-tuples of holomorphic sections of Lp⊗E. Using these results and a sequence of suitable meromorphic transforms associated to the degeneracy sets, we prove the almost sure convergence of these currents as p→∞. This talk is based on joint work with Turgay Bayraktar, Bingxiao Liu and George Marinescu.\n\nhttps://indico.math.cnrs.fr/event/16259/