Numerical flatness and principal bundles on Fujiki manifolds

TL;DR

This paper establishes equivalences between numerical flatness of principal bundle adjoint bundles, nefness of certain line bundles, and degree inequalities on Fujiki manifolds, extending concepts of flatness in complex geometry.

Contribution

It proves the equivalence of three conditions related to principal G-bundles on Fujiki manifolds, linking numerical flatness, nefness, and degree inequalities, thus advancing the understanding of flatness criteria.

Findings

Numerical flatness of ad(E_G) is equivalent to nefness of a top wedge line bundle.

Degree inequalities hold for all reductions over Kähler spaces.

The results unify flatness conditions for principal bundles on Fujiki manifolds.

Abstract

Let $M$ be a compact connected Fujiki manifold, $G$ a semisimple affine algebraic group over $\mathbb C$ with one simple factor and $P$ a fixed proper parabolic subgroup of $G$. For a holomorphic principal $G$--bundle $E_G$ over $M$, let ${\mathcal E}_P$ be the holomorphic principal $P$-bundle $E_G\rightarrow E_G/P$ given by the quotient map. We prove that the following three statements are equivalent: (1) ${\rm ad}(E_G)$ is numerically flat, (2) the holomorphic line bundle $\bigwedge^{\rm top} {\rm ad}({\mathcal E}_P)^*$ is nef, and (3) for every reduced irreducible compact complex analytic space $Z$ with a K\"ahler form $\omega$, holomorphic map $\gamma : Z \rightarrow M$, and holomorphic reduction of structure group $E_P \subset \gamma^*E_G$ to $P$, the inequality ${\rm degree}({\rm ad}(E_P)) \leq 0$ holds.