Number variance of random zeros on complex manifolds
| dc.creator | Shiffman, Bernard | |
| dc.creator | Zelditch, Steve | |
| dc.date | 2006-08-30 | |
| dc.date | 2008-11-26 | |
| dc.date.accessioned | 2026-07-07T12:21:39Z | |
| dc.date.available | 2026-07-07T12:21:39Z | |
| dc.description | 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 $U \subset C^m$ with smooth boundary is asymptotic to $N^{m-1/2} ν_{mm} Vol(\partial U)$, 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. | |
| dc.description | Some computations are simplified and positivity of the coefficient of the leading term in the variance formula is shown for all codimensions. This article is a follow-up to math/0512652, which dealt with zero sets of codimension one. The original posting (v1) also contains results on smooth linear statistics and on random holomorphic functions on noncompact manifolds | |
| dc.identifier | https://arxiv.org/abs/math/0608743 | |
| dc.identifier | http://arxiv.org/abs/math/0608743 | |
| dc.identifier | Geom. Funct. Anal. 18 (2008), 1422-1475. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/213403 | |
| dc.subject | Complex Variables | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Probability | |
| dc.title | Number variance of random zeros on complex manifolds | |
| dc.type | text |