Probability Measures Associated to Geodesics in the Space of Kähler Metrics
نویسنده
چکیده
We associate certain probability measures on R to geodesics in the space HL of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H(X, kL). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in HL 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.
منابع مشابه
Bergman Metrics and Geodesics in the Space of Kähler Metrics on Toric Varieties
Geodesics on the infinite dimensional symmetric space H of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in X×A, where A ⊂ C is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces GC/G. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Ampère geodesics c...
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