Title: Zeros of random holomorphic sections of big line bundles with continuous metrics Show Chinese title

Title: 大线丛的随机全纯截面的零点与连续度量

Abstract: Let $X$ be a compact normal complex space, $L$ be a big holomorphic line bundle on $X$ and $h$ be a continuous Hermitian metric on $L$. We consider the spaces of holomorphic sections $H^0(X, L^{\otimes p})$ endowed with the inner product induced by $h^{\otimes p}$ and a volume form on $X$, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current $c_1(L,h_{\mathrm{eq}})$ of the equilibrium metric $h_{\mathrm{eq}}$ associated to $h$. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to $c_1(L,h_{\mathrm{eq}})$, for very general classes of probability measures on $H^0(X, L^{\otimes p})$. Show Chinese abstract

Abstract: 设$X$为一个紧致的正规复空间，$L$为$X$上的一个大全纯线丛，$h$为$L$上的一个连续埃尔米特度量。 我们考虑由$h^{\otimes p}$诱导的内积和$X$上的体积形式赋予的全纯截面空间$H^0(X, L^{\otimes p})$，并证明相应的归一化Fubini-Study当前序列弱收敛到与$h$相关的平衡度量$h_{\mathrm{eq}}$的曲率当前$c_1(L,h_{\mathrm{eq}})$。 我们还证明了，随机全纯截面序列沿零除子的归一化电流几乎必然收敛到$c_1(L,h_{\mathrm{eq}})$，对于$H^0(X, L^{\otimes p})$上非常一般的概率测度类。