• Corpus ID: 119297784

# $C^{1,1}$ regularity of geodesics of singular K\"{a}hler metrics

@article{Chu2019C11RO,
title={\$C^\{1,1\}\$ regularity of geodesics of singular K\"\{a\}hler metrics},
author={Jianchun Chu and Nicholas McCleerey},
journal={arXiv: Differential Geometry},
year={2019}
}
• Published 7 January 2019
• Mathematics
• arXiv: Differential Geometry
We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on K\"ahler manifolds away from the non-K\"ahler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of K\"ahler metrics on compact K\"ahler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge-Amp\ere equation that does not require strict positivity of the reference form near the…
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## References

SHOWING 1-10 OF 53 REFERENCES
Morse theory and geodesics in the space of Kähler metrics
Given a compact K\"ahler manifold $(X,\omega_0)$ let $\mathcal H_{0}$ be the set of K\"ahler forms cohomologous to $\omega_0$. As observed by Mabuchi \cite{m}, this space has the structure of an
On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold
Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the
The Space of K\"ahler metrics (II)
• Mathematics
• 2001
This paper, the second of a series, deals with the function space of all smooth K\"ahler metrics in any given closed complex manifold $M$ in a fixed cohomology class. The previous result of the
Geometry and topology of the space of Kähler metrics on singular varieties
• Mathematics
Compositio Mathematica
• 2018
Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space
WEAK GEODESIC RAYS IN THE SPACE OF KÄHLER POTENTIALS AND THE CLASS ${\mathcal{E}}(X,\unicode[STIX]{x1D714})$
• T. Darvas
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2015
Suppose that $(X,\unicode[STIX]{x1D714})$ is a compact Kähler manifold. In the present work we propose a construction for weak geodesic rays in the space of Kähler potentials that is tied together
Metric geometry of normal K\"ahler spaces, energy properness, and existence of canonical metrics
Let $(X,\omega)$ be a compact normal K\"ahler space, with Hodge metric $\omega$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays
• Mathematics
• 2017
ABSTRACT We prove a C1,1 estimate for solutions of complex Monge–Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens
On the Regularity of Geodesics in the Space of Kähler Metrics
• Mathematics
• 2016
We prove that any two Kähler potentials on a compact Kähler manifold can be connected by a geodesic segment of $$C^{1,1}$$C1,1 regularity. This follows from an a priori interior real Hessian bound
Envelopes with Prescribed Singularities
We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal $$C^{1,1}$$C1,1 regularity on a
On the singularity type of full mass currents in big cohomology classes
• Mathematics
Compositio Mathematica
• 2017
Let $X$ be a compact Kähler manifold and $\{\unicode[STIX]{x1D703}\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open