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Kuang-Ru Wu

Publications and source records attributed to Kuang-Ru Wu.

12 recordsLinked to original sources
arXiv2026

RC-positivity of complex surfaces

Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.

math.DG↗
arXiv2026

Uniform weak RC-positivity and rational connectedness

In this paper, we show that if the holomorphic tangent bundle $TX$ of a compact Kähler manifold $X$ is uniformly weakly RC-positive, then $X$ is projective and rationally connected. This result is previously established by Xiaokui Yang under the stronger assumption that $TX$ is uniformly RC-positive. The result we obtain is, in fact, more general. If a holomorphic vector bundle $E$ is uniformly weakly RC-positive, then $E$ admits a Hermitian metric whose mean curvature is positive. A quasi-positive version is also proved in this paper.

math.DG↗
arXiv2025

Uniform RC-positivity of direct image bundles

The concept of RC-positivity and uniform RC-positivity is introduced by Xiaokui Yang to solve a conjecture of Yau on projectivity and rational connectedness of a compact Kähler manifold with positive holomorphic sectional curvature. Some main theorems in Yang's proof hold under a weaker condition called weak RC-positivity. It is therefore natural to ask if (uniform) weak RC-positivity implies (uniform) RC-positivity. Another motivation for studying this problem is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. In this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$.

math.DG↗
arXiv2025

A potential theory for the Wess--Zumino--Witten equation in the space of Kähler potentials

We develop a potential theory for the Wess--Zumino--Witten (WZW) equation in the space of Kähler potentials which is parallel to the potential theory for the Hermitian--Yang--Mills equation. A concept called $ω$-harmonicity on graphs is introduced which characterizes the WZW equation. We also show that, with respect to a Banach--Mazur type distance function, the distance between two solutions of the WZW equation is subharmonic. The harmonic map into the space of Kähler potentials, as a special case of the WZW equation, is also investigated. In particular, we show the solvability of the Dirichlet problem for the harmonic map, and the approximation/quantization by its finite dimensional counterparts.

math.DG↗
arXiv2025

Mean curvature of direct image bundles

Let $E\to X$ be a vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. It is known that if the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ is positive, then $E\otimes \det E$ is Nakano positive by the work of Berndtsson. In this paper, we give a subharmonic analogue. Let $p:P(E^*)\to X$ be the projection and $α$ be a Kähler form on $X$. If the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $Θ$ positive on every fiber and $Θ^r\wedge p^*α^{n-1}> 0$, then $E\otimes \det E$ carries a Hermitian metric whose mean curvature is positive. As an application, we show that the following subharmonic analogue of the Griffiths conjecture is true: if the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $Θ$ positive on every fiber and $Θ^r\wedge p^*α^{n-1}> 0$, then $E$ carries a Hermitian metric with positive mean curvature.

math.DG↗
arXiv2023

Positively curved Finsler metrics on vector bundles III

The goal of the paper is to extend results about ample or Griffiths positive vector bundles to Kobayashi positive vector bundles. In particular, we show that the quotient bundle of a Kobayashi positive vector bundle is Kobayashi positive, and the tensor product of two Kobayashi positive vector bundles is Kobayashi positive. These results strengthen the conjectural equivalences between ampleness, Griffiths positivity, and Kobayashi positivity. The proofs rely on the convexity of Kobayashi positive Finsler metrics and the duality for convex Finsler metrics.

math.DG↗
arXiv2022

Positively curved Finsler metrics on vector bundles II

We show that if $E$ is an ample vector bundle of rank at least two with some curvature bound on $O_{P(E^*)}(1)$, then $E^*\otimes \det E$ is Kobayashi positive. The proof relies on comparing the curvature of $(\det E^*)^k$ and $S^kE$ for large $k$ and using duality of convex Finsler metrics. Following the same thread of thought, we show if $E$ is ample with similar curvature bounds on $O_{P(E^*)}(1)$ and $O_{P(E\otimes \det E^*)}(1)$, then $E$ is Kobayashi positive. With additional assumptions, we can furthermore show that $E^*\otimes \det E$ and $E$ are Griffiths positive.

math.DG↗
arXiv2022

Griffiths extremality, interpolation of norms, and Kähler quantization

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge--Ampère type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kähler geometry, related to the construction of flat maps for the Mabuchi metric.

math.DG↗
arXiv2021

Positively curved Finsler metrics on vector bundles

We construct a convex and strongly pseudoconvex Kobayashi positive Finsler metric on a vector bundle $E$ under the assumption that the symmetric power of the dual $S^kE^*$ has a Griffiths negative $L^2$-metric for some $k$. The proof relies on the negativity of direct image bundles and the Minkowski inequality for norms. As a corollary, we show that given a strongly pseudoconvex Kobayashi positive Finsler metric, one can upgrade to a \textit{convex} Finsler metric with the same property. We also give an extremal characterization of Kobayashi curvature for Finsler metrics.

math.CV↗
arXiv2018

A Dirichlet problem in noncommutative potential theory

We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.

math.CV↗