Kähler hyperbolic manifold

In complex differential geometry, a branch of mathematics , Kähler manifolds are called Kähler hyperbolic if the elevated Kähler form of the universal superposition is the differential of a restricted differential form. ${\displaystyle (M,\omega )}$ ${\displaystyle {\widetilde {\omega }}}$ ${\displaystyle {\widetilde {M}}}$

Examples

Theorem (Gromow): A closed Kähler manifold with negative section curvature is Kähler hyperbolic. Every Kähler manifold that is homotopy-equivalent to a closed Riemannian manifold of negative intersection curvature is Kähler hyperbolic.

Further sufficient conditions for Kähler hyperbolicity of Kähler manifolds:

• ${\displaystyle \pi _{1}M}$is hyperbolic and ${\displaystyle \pi _{2}M=0}$
• ${\displaystyle M}$ is the submanifold of a Kähler hyperbolic manifold
• ${\displaystyle {\widetilde {M}}}$is a Hermitian symmetric space of non-compact type with no Euclidean factor

McMullen proved that the Teichmüller space is Kähler hyperbolic.

Applications

Gromow proved that the Hopf conjecture is correct for Kähler hyperbolic manifolds . This means that for Riemannian 2n manifolds with negative sectional curvature the inequality

${\displaystyle (-1)^{n}\chi (M)>0}$

applies. Here is the Euler characteristic . ${\displaystyle \chi (M)}$

Other notions of hyperbolicity

Every Kähler hyperbolic manifold is Kobayashi hyperbolic, i.e. H. every holomorphic map is constant. ${\displaystyle X}$ ${\displaystyle \mathbb {C} \to X}$

literature

• Werner Ballmann : Lectures on Kähler manifolds. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zurich, 2006. ISBN 978-3-03719-025-8 pdf (Chapter 8)
• Gromov, M .: Kähler hyperbolicity and L ² -Hodge theory. J. Differential Geom. 33 (1991) no. 1, 263-292. pdf

Individual evidence

1. ^ McMullen, Curtis T .: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. of Math. (2) 151 (2000), no. 1, 327-357.