Loop space of a Kähler manifold

TL;DR

The paper shows that the loop space $LX$ of a complex Kähler manifold inherits a Kähler structure via a natural $L^2$-type metric, with the induced form $\Omega$ closed when the base form $d\omega$ vanishes. It analyzes two infinite-dimensional projective-space models to illustrate metric behavior: $L{\mathbb P}^1$ is complete but unbounded, while ${\mathbb P}(l^2)$ is complete and bounded under the Fubini-Study metric. A Levi-Civita connection on $LX$ is constructed by lifting the base Levi-Civita connection, leading to the geodesic equation on $LX$ and a leafwise interpretation: geodesics in $LX$ correspond to pointwise geodesics in $X$. These results establish a coherent geometric framework for the loop-space metric and its geodesics, with implications for infinite-dimensional Kähler geometry and leafwise geodesic construction.

Abstract

We prove that the loop space of a Kähler manifold inherits a Kähler structure. Then we prove that equipped with this natural metric the loop space is complete and unbounded. Additionally, we show that a geodesic on the loop space can be constructed by piecing together geodesics from each individual leaf.

Theorems & Definitions (12)

• Theorem 1
• Corollary 1
• Theorem 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.1
• Corollary 3.1
• Proposition 5.1