Kähler度量空间中与测地线有关的概率测度
Probability measures related to geodesics in the space of Kähler metrics
摘要：
我们将$\R$上的某些概率测度与线丛$L$上的正曲度量空间$\H_L$中的测地线以及$H^0(X, kL)$上的厄米范数的有限维对称空间中的测地线联系起来。证明了当$k$趋于无穷大时，有限维空间的测度弱收敛于$\H_L$中测地线的测度.二阶矩的收敛性暗示了Chen和Sun最近关于各自空间中测地距离的结果，而一阶矩的收敛性给出了Donaldson的$Z$ -泛函收敛于Aubin-Yau能量.我们还包括一个结果逼近无穷维测地线的Bergman内核，推广了工作的Phong和Sturm。
英文摘要：
We associate certain probability measures on $\R$ to geodesics in the space $\H_L$ of positively curved metrics on a line bundle $L$, and to geodesics in the finite dimensional symmetric space of hermitian norms on $H^0(X, kL)$. We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in $\H_L$ as $k$ goes to infinity. The convergence of second order moments implies a recent result of Chen and Sun on geodesic distances in the respective spaces, while the convergence of first order moments gives convergence of Donaldson's $Z$-functional to the Aubin-Yau energy. We also include a result on approximation of infinite dimensional geodesics by Bergman kernels which generalizes work of Phong and Sturm.
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