Expected local topology of random complex submanifolds

Abstract

Let n ≥ 2 n\geq 2 and r ∈ { 1 , ⋯ , n − 1 } r\in \{1, \cdots , n-1\} be integers, M M be a compact smooth Kähler manifold of complex dimension n n , E E be a holomorphic vector bundle with complex rank r r and equipped with a Hermitian metric h E h_E , and L L be an ample holomorphic line bundle over M M equipped with a metric h h with positive curvature form. For any d ∈ N d\in \mathbb N large enough, we equip the space of holomorphic sections H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) with the natural Gaussian measure associated to h E h_E , h h and its curvature form. Let U ⊂ M U\subset M be an open subset with smooth boundary. We prove that the average of the ( n − r ) (n-r) -th Betti number of the vanishing locus in U U of a random section s s of H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) is asymptotic to ( n − 1 r − 1 ) d n ∫ U c 1 ( L ) n {n-1 \choose r-1} d^n\int _U c_1(L)^n for large d d . On the other hand, the average of the other Betti numbers is o ( d n ) o(d^n) . The first asymptotic recovers the classical deterministic global algebraic computation. Moreover, such a discrepancy in the order of growth of these averages is new and contrasts with all known other smooth Gaussian models, in particular the real algebraic one. We prove a similar result for the affine complex Bargmann-Fock model.