We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers L N of a positive holomorphic Hermitian line bundle L over a compact complex manifold M . Our first result concerns 'random' sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {S N j } of H 0 (M, L N ), we show that for almost every sequence {S N j }, the associated sequence of zero currents 1 N Z S N j tends to the curvature form ω of L. Thus, the zeros of a sequence of sections s N ∈ H 0 (M, L N ) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {S N j } of H 0 (M, L N ) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

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We study the limit as N → ∞ of the correlations between simultaneous zeros of random sections of the powers L N of a positive holomorphic line bundle L over a compact complex manifold M , when distances are rescaled so that the average density of zeros is independent of N . We show that the limit correlation is independent of the line bundle and depends only on the dimension of M and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay's limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.

In their work on symplectic manifolds, Donaldson and Auroux use analogues of holomorphic sections of an ample line bundle L over a symplectic manifold M to create symplectically embedded zero sections and almost holomorphic maps to various spaces. Their analogues were termed 'asymptotically holomorphic' sequences fs N g of sections of L N . We study another analogue H 0 J ðM; L N Þ of holomorphic sections, which we call 'almost-holomorphic' sections, following a method introduced earlier by Boutet de Monvel-Guillemin [BoGu] in a general setting of symplectic cones. By definition, sections in H 0 J ðM; L N Þ lie in the range of a Szegö projector P N . Starting almost from scratch, and only using almost-complex geometry, we construct a simple parametrix for P N of precisely the same type as the Boutet de Monvel-Sjö strand parametrix in the holomorphic case [BoSj]. We then show that P N ðx; yÞ has precisely the same scaling asymptotics as does the holomorphic Szegö kernel as analyzed in [BSZ1]. The scaling asymptotics imply more or less immediately a number of analogues of well-known results in the holomorphic case, e.g. a Kodaira embedding theorem and a Tian almost-isometry theorem. We also explain how to modify Donaldson's constructions to prove existence of quantitatively transverse sections in H 0 J ðM; L N Þ.

We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set U ⊂ C m with smooth boundary is asymptotic to N m−1/2 ν mm Vol(∂U ), where ν mm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on C m . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N -th power of any positive line bundle over any m-dimensional compact Kähler manifold.