Hermitian-Poisson Metrics Overview

• Hermitian-Poisson metrics are Hermitian metrics on complex vector bundles over Gauduchon manifolds where a curvature trace yields a scalar endomorphism, distinct from classical Hermitian-Einstein metrics.
• They arise in both non-compact and dimension-reduction settings, linking projectively flat bundles to non-abelian Hodge theory and providing insights into stability and gauge theoretic applications.
• Analytic tools such as heat flow, continuity methods, and L2 estimates are employed to establish existence, uniqueness, and semisimplicity criteria in these non-Kähler frameworks.

Searching arXiv for the cited papers and closely related work on Hermitian-Poisson / Poisson metrics. arXiv search query: "Hermitian-Poisson metrics projectively flat non-compact Gauduchon manifolds" Looking up arXiv metadata for the main paper via the arXiv API. $\sqrt{-1}\Lambda_\omega G_H$94 $\sqrt{-1}\Lambda_\omega G_H$95 Hermitian-Poisson metrics are Hermitian metrics on flat or projectively flat complex vector bundles for which an appropriate curvature trace is a scalar endomorphism. In the non-Kähler extension of non-abelian Hodge theory, tailored to projectively flat complex vector bundles over Gauduchon manifolds, the defining equation is expressed in terms of the “complex curvature” $G_H=(D_H'')^2$ and its contraction $\sqrt{-1}\Lambda_\omega G_H$ (Geng et al., 15 Jul 2025). On non-compact curves, the same terminology is used for metrics satisfying an affine divergence equation $K(H)=c\,I$, where $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$; in that setting the equation is a deformation of Simpson’s harmonic metric equation and arises by dimension reduction of the Hermitian–Yang–Mills equation from K3 surfaces in the large complex structure limit (Collins et al., 2014).

1. Definitions and geometric setting

On a complex Hermitian manifold $(X,\omega)$ of complex dimension $n$, the metric is Gauduchon if

$\partial\bar\partial(\omega^{n-1})=0.$

The non-compact framework considered for Hermitian-Poisson metrics assumes that $\omega$ is Gauduchon and satisfies additional analytic hypotheses, including finite volume and the existence of an exhaustion function with bounded $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$ (Geng et al., 15 Jul 2025).

Let $(E,D)$ be a complex vector bundle of rank $\sqrt{-1}\Lambda_\omega G_H$0 over $\sqrt{-1}\Lambda_\omega G_H$1 equipped with a complex connection $\sqrt{-1}\Lambda_\omega G_H$2. For any Hermitian metric $\sqrt{-1}\Lambda_\omega G_H$3 on $\sqrt{-1}\Lambda_\omega G_H$4, there is a unique decomposition

$\sqrt{-1}\Lambda_\omega G_H$5

where $\sqrt{-1}\Lambda_\omega G_H$6 is unitary and $\sqrt{-1}\Lambda_\omega G_H$7 is self-adjoint with respect to $\sqrt{-1}\Lambda_\omega G_H$8. Writing the $\sqrt{-1}\Lambda_\omega G_H$9 and $K(H)=c\,I$0 parts of $K(H)=c\,I$1 as $K(H)=c\,I$2 and $K(H)=c\,I$3, one defines

$K(H)=c\,I$4

and the associated complex curvature

$K(H)=c\,I$5

The contraction operator $K(H)=c\,I$6 is then used to form the Poisson trace $K(H)=c\,I$7 (Geng et al., 15 Jul 2025).

A connection $K(H)=c\,I$8 is projectively flat if its curvature is a scalar multiple of the identity: $K(H)=c\,I$9 for some real $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$0-form $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$1. In terms of the decomposition $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$2, this implies

$K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$3

together with the component relations listed in the projectively flat system (Geng et al., 15 Jul 2025).

In the curve case, the same geometric pattern is expressed differently. Given a flat bundle $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$4 over a punctured Riemann surface, a Hermitian metric $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$5 determines a splitting

$K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$6

where $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$7 is metric-unitary and $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$8 is an $K(H)=-\,\star\,\nabla\,\star\,\Psi(H)$9-self-adjoint endomorphism-valued $(X,\omega)$0-form. The dual connection is

$(X,\omega)$1

In a flat frame,

$(X,\omega)$2

The corresponding endomorphism

$(X,\omega)$3

is the affine “trace of curvature” entering the curve-level Poisson metric equation (Collins et al., 2014).

2. Hermitian-Poisson equations and their relation to Hermitian-Einstein theory

Following Pan–Zhang–Zhang and the convention adopted in the Gauduchon-manifold setting, a Hermitian metric $(X,\omega)$4 on $(X,\omega)$5 is called a Hermitian-Poisson metric if

$(X,\omega)$6

for some real constant $(X,\omega)$7. Equivalently, the $(X,\omega)$8-contraction of $(X,\omega)$9 against $n$0 is a scalar endomorphism of $n$1 (Geng et al., 15 Jul 2025).

For projectively flat bundles, the role of $n$2 is essential. The Hermitian-Poisson condition constrains the trace of the complex curvature $n$3 rather than the full Chern curvature $n$4. This is the principal distinction from the Hermitian-Einstein condition

$n$5

In the Kähler case, Donaldson’s functional and Simpson’s heat flow provide variational control for Hermitian-Einstein metrics and Higgs bundles. In Gauduchon, non-Kähler settings, Donaldson’s functional is unavailable, and the analysis is reorganized around $n$6, its Poisson trace, and analytic substitutes for the missing variational structure (Geng et al., 15 Jul 2025).

On non-compact curves, “Poisson metric” and “Hermitian–Poisson metric” are synonymous. The defining equation is

$n$7

On curves this coincides with

$n$8

When $n$9, the equation becomes Simpson’s harmonic metric equation, so the Poisson equation is a deformation by a constant Poisson term (Collins et al., 2014).

The curve equation also admits a local affine form: $\partial\bar\partial(\omega^{n-1})=0.$0 This is obtained by dimension reduction of the Hermitian–Yang–Mills trace equation on a holomorphic bundle over $\partial\bar\partial(\omega^{n-1})=0.$1, and in the large complex structure limit of an elliptic K3 surface the same reduction yields the Poisson PDE on the base curve. In that sense, the Poisson metric equation is described as the affine analogue of the Hermitian–Yang–Mills equation under SYZ-type dimension reduction (Collins et al., 2014).

A common misconception is to identify Hermitian-Poisson metrics with Hermitian-Einstein metrics. The two notions coincide neither analytically nor geometrically in general. In the Gauduchon-manifold setting, the equation is built from $\partial\bar\partial(\omega^{n-1})=0.$2, while in the affine curve setting it is built from the divergence of $\partial\bar\partial(\omega^{n-1})=0.$3. This suggests that “Poisson” refers not to a single universal curvature tensor but to a family of scalar-trace conditions adapted to flat or projectively flat data outside the standard Kähler framework.

3. Existence correspondences and semisimplicity criteria

The principal non-compact correspondence for projectively flat bundles over Gauduchon manifolds is an existence-and-semisimplicity theorem under explicit analytic assumptions (Geng et al., 15 Jul 2025). Let $\partial\bar\partial(\omega^{n-1})=0.$4 be a non-compact Gauduchon manifold satisfying:

• finite volume $\partial\bar\partial(\omega^{n-1})=0.$5;
• existence of an exhaustion function $\partial\bar\partial(\omega^{n-1})=0.$6 with bounded $\partial\bar\partial(\omega^{n-1})=0.$7;
• an assumption controlling sup norms from integrated inequalities;
• $\partial\bar\partial(\omega^{n-1})=0.$8;
• a background Hermitian metric $\partial\bar\partial(\omega^{n-1})=0.$9 with $\omega$0.

Under these hypotheses, if $\omega$1 is simple and projectively flat, there exists a Hermitian metric $\omega$2 such that $\omega$3 and $\omega$4 are mutually bounded,

$\omega$5

and

$\omega$6

Moreover, if $\omega$7, then $\omega$8 (Geng et al., 15 Jul 2025).

The converse statement requires additional geometry. If $\omega$9 is balanced and $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$0 admits a Hermitian-Poisson metric $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$1 with $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$2, then $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$3 is semi-simple. Here “simple” means that $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$4 admits no proper $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$5-invariant subbundle, while “semi-simple” means a direct sum of simple $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$6-invariant subbundles. Thus, under the balanced condition and $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$7 control, existence of a Hermitian-Poisson metric characterizes semi-simplicity (Geng et al., 15 Jul 2025).

The curve case gives an analogous but differently formulated correspondence. Let $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$8 be a punctured compact Riemann surface endowed with a smooth Kähler metric $\sqrt{-1}\Lambda_\omega\partial\bar\partial\phi$9 of finite volume. For a flat vector bundle with regular singularities and a parabolic structure $(E,D)$0, the main theorem states that the bundle admits a conformally strongly tamed Poisson metric if and only if it is slope polystable. If $(E,D)$1 for some $(E,D)$2, then the metric is strongly tamed. Any such metric is unique up to multiplication by a positive constant (Collins et al., 2014).

The stability mechanism on curves is encoded by a Chern–Weil formula. If $(E,D)$3 is a flat subbundle with orthogonal projection $(E,D)$4 and second fundamental form $(E,D)$5, then

$(E,D)$6

If $(E,D)$7 is Poisson, this yields $(E,D)$8, with equality if and only if $(E,D)$9, so existence of a Poisson metric implies slope polystability (Collins et al., 2014).

4. Analytic framework: heat flow, continuity method, and substitute functionals

The non-compact Gauduchon theory combines heat flow techniques and continuity methods. For $\sqrt{-1}\Lambda_\omega G_H$00, the perturbed equation is

$\sqrt{-1}\Lambda_\omega G_H$01

Its associated heat flow is

$\sqrt{-1}\Lambda_\omega G_H$02

Writing $\sqrt{-1}\Lambda_\omega G_H$03 and $\sqrt{-1}\Lambda_\omega G_H$04, the flow satisfies

$\sqrt{-1}\Lambda_\omega G_H$05

These monotonicity relations imply uniform bounds by the maximum principle (Geng et al., 15 Jul 2025).

A second fundamental tool is Donaldson’s distance

$\sqrt{-1}\Lambda_\omega G_H$06

For two solutions with the same initial data,

$\sqrt{-1}\Lambda_\omega G_H$07

which implies uniqueness and $\sqrt{-1}\Lambda_\omega G_H$08 convergence. The flow also yields the estimate

$\sqrt{-1}\Lambda_\omega G_H$09

and local $\sqrt{-1}\Lambda_\omega G_H$10 bounds for $\sqrt{-1}\Lambda_\omega G_H$11 on compact subsets (Geng et al., 15 Jul 2025).

Because Donaldson’s functional is not available in the Gauduchon setting, the paper replaces it by a central integral identity. If $\sqrt{-1}\Lambda_\omega G_H$12, then

$\sqrt{-1}\Lambda_\omega G_H$13

where

$\sqrt{-1}\Lambda_\omega G_H$14

and

$\sqrt{-1}\Lambda_\omega G_H$15

This identity is crucial for global $\sqrt{-1}\Lambda_\omega G_H$16 and $\sqrt{-1}\Lambda_\omega G_H$17 estimates, and for the passage $\sqrt{-1}\Lambda_\omega G_H$18 to an actual Hermitian-Poisson metric (Geng et al., 15 Jul 2025).

A related Bochner-type identity is

$\sqrt{-1}\Lambda_\omega G_H$19

and it drives the evolution of $\sqrt{-1}\Lambda_\omega G_H$20. On non-compact manifolds, the analysis is localized to exhausting domains $\sqrt{-1}\Lambda_\omega G_H$21, with Dirichlet boundary condition $\sqrt{-1}\Lambda_\omega G_H$22, and then passed to the limit using uniform $\sqrt{-1}\Lambda_\omega G_H$23 and local $\sqrt{-1}\Lambda_\omega G_H$24 estimates (Geng et al., 15 Jul 2025).

The curve theory uses a parallel, though not identical, analytic scheme. On compact submanifolds $\sqrt{-1}\Lambda_\omega G_H$25 with boundary, one solves Donaldson’s heat flow

$\sqrt{-1}\Lambda_\omega G_H$26

with prescribed boundary value $\sqrt{-1}\Lambda_\omega G_H$27. The curvature satisfies $\sqrt{-1}\Lambda_\omega G_H$28, decays exponentially, and the flow converges to a stationary solution $\sqrt{-1}\Lambda_\omega G_H$29 with $\sqrt{-1}\Lambda_\omega G_H$30. If $\sqrt{-1}\Lambda_\omega G_H$31 were unbounded, one constructs a weak $\sqrt{-1}\Lambda_\omega G_H$32-projection onto a proper flat subbundle, contradicting stability (Collins et al., 2014).

5. Parabolic structures, asymptotics, and local models on non-compact curves

On a punctured Riemann surface $\sqrt{-1}\Lambda_\omega G_H$33, the curve theory incorporates regular singularities and parabolic structures. Near each puncture, after a suitable meromorphic gauge, a regular singular flat connection has local form

$\sqrt{-1}\Lambda_\omega G_H$34

or equivalently in polar coordinates,

$\sqrt{-1}\Lambda_\omega G_H$35

A parabolic structure assigns real weights $\sqrt{-1}\Lambda_\omega G_H$36 to the indecomposable summands $\sqrt{-1}\Lambda_\omega G_H$37 in the local decomposition $\sqrt{-1}\Lambda_\omega G_H$38 (Collins et al., 2014).

The parabolic degree is

$\sqrt{-1}\Lambda_\omega G_H$39

and the parabolic slope is $\sqrt{-1}\Lambda_\omega G_H$40. One says $\sqrt{-1}\Lambda_\omega G_H$41 is slope stable if $\sqrt{-1}\Lambda_\omega G_H$42 for every proper flat subbundle $\sqrt{-1}\Lambda_\omega G_H$43, semistable if $\sqrt{-1}\Lambda_\omega G_H$44, and polystable if it is a direct sum of stable flat subbundles of the same slope (Collins et al., 2014).

The asymptotic side of the theory is encoded by tame and strongly tame metrics. A metric $\sqrt{-1}\Lambda_\omega G_H$45 is tamed by $\sqrt{-1}\Lambda_\omega G_H$46 if it satisfies integrability of $\sqrt{-1}\Lambda_\omega G_H$47 and radial asymptotics of $\sqrt{-1}\Lambda_\omega G_H$48 on local invariant flat subbundles: $\sqrt{-1}\Lambda_\omega G_H$49 For strong tameness, one adds precise norm growth of flat sections and a fuller asymptotic expansion of $\sqrt{-1}\Lambda_\omega G_H$50, including suppressed nilpotent terms of size $\sqrt{-1}\Lambda_\omega G_H$51 in the angular component (Collins et al., 2014).

The local model solutions are explicit. If near a puncture

$\sqrt{-1}\Lambda_\omega G_H$52

with $\sqrt{-1}\Lambda_\omega G_H$53 a single Jordan block of size $\sqrt{-1}\Lambda_\omega G_H$54, then the diagonal metric

$\sqrt{-1}\Lambda_\omega G_H$55

solves $\sqrt{-1}\Lambda_\omega G_H$56 and is strongly tame (Collins et al., 2014).

These local models are glued and then conformally corrected by solving

$\sqrt{-1}\Lambda_\omega G_H$57

to obtain a strongly conformally tame metric solving $\sqrt{-1}\Lambda_\omega G_H$58 near each puncture. If $\sqrt{-1}\Lambda_\omega G_H$59 for some $\sqrt{-1}\Lambda_\omega G_H$60, the metric is strongly tame without further conformal correction (Collins et al., 2014).

The constant $\sqrt{-1}\Lambda_\omega G_H$61 is determined by the parabolic degree. For any tame metric $\sqrt{-1}\Lambda_\omega G_H$62,

$\sqrt{-1}\Lambda_\omega G_H$63

and therefore a Poisson metric satisfies

$\sqrt{-1}\Lambda_\omega G_H$64

This normalization matches the affine constant appearing in the dimension-reduced Hermitian–Yang–Mills equation (Collins et al., 2014).

6. Splitting phenomena, examples, non-examples, and open directions

A major structural consequence of the Gauduchon-manifold theory is that Hermitian-Poisson metrics detect semi-simplicity. One mechanism uses normalized logarithms $\sqrt{-1}\Lambda_\omega G_H$65, spectral analysis in the style of Simpson, and step functions $\sqrt{-1}\Lambda_\omega G_H$66 applied to the limit $\sqrt{-1}\Lambda_\omega G_H$67. This yields self-adjoint $\sqrt{-1}\Lambda_\omega G_H$68 projections $\sqrt{-1}\Lambda_\omega G_H$69 satisfying

$\sqrt{-1}\Lambda_\omega G_H$70

Each $\sqrt{-1}\Lambda_\omega G_H$71 projects onto a $\sqrt{-1}\Lambda_\omega G_H$72-invariant subbundle. In the balanced case, a block decomposition

$\sqrt{-1}\Lambda_\omega G_H$73

together with the Hermitian-Poisson equation implies

$\sqrt{-1}\Lambda_\omega G_H$74

so the bundle splits orthogonally into a direct sum of $\sqrt{-1}\Lambda_\omega G_H$75-invariant subbundles (Geng et al., 15 Jul 2025).

Several examples and non-examples are recorded. For the trivial bundle $\sqrt{-1}\Lambda_\omega G_H$76 with $\sqrt{-1}\Lambda_\omega G_H$77 unitary and projectively flat, the Hermitian-Poisson equation reduces to $\sqrt{-1}\Lambda_\omega G_H$78, and a constant metric with $\sqrt{-1}\Lambda_\omega G_H$79 satisfies the equation with $\sqrt{-1}\Lambda_\omega G_H$80. If $\sqrt{-1}\Lambda_\omega G_H$81 is flat, then projectively flatness holds trivially, and under the stated assumptions one obtains Hermitian-Poisson metrics with normalization determined by $\sqrt{-1}\Lambda_\omega G_H$82 and $\sqrt{-1}\Lambda_\omega G_H$83. Semi-simple direct sums of simple projectively flat factors admit Hermitian-Poisson metrics by solving on each factor, subject to the normalization and trace constraints (Geng et al., 15 Jul 2025).

In the curve setting, illustrative cases include a rank-two trivial bundle over $\sqrt{-1}\Lambda_\omega G_H$84 with connection

$\sqrt{-1}\Lambda_\omega G_H$85

for which the only invariant subbundle is $\sqrt{-1}\Lambda_\omega G_H$86, and strongly tame metrics with logarithmic growth such as

$\sqrt{-1}\Lambda_\omega G_H$87

near a puncture (Collins et al., 2014).

The main non-examples are analytic rather than algebraic. If the non-compact Gauduchon manifold fails the hypotheses of finite volume, controlled exhaustion, the $\sqrt{-1}\Lambda_\omega G_H$88 condition $\sqrt{-1}\Lambda_\omega G_H$89, or the integrated inequality encoded in Assumption 3, then the analytic machinery may fail. Likewise, without balanced $\sqrt{-1}\Lambda_\omega G_H$90 and $\sqrt{-1}\Lambda_\omega G_H$91 control on $\sqrt{-1}\Lambda_\omega G_H$92, the converse implication from existence of a Hermitian-Poisson metric to semi-simplicity may not hold (Geng et al., 15 Jul 2025).

The current literature also identifies several open directions. These include removing the balanced hypothesis in the converse, weakening the $\sqrt{-1}\Lambda_\omega G_H$93 assumptions, developing moduli-theoretic interpretations for Hermitian-Poisson metrics in non-Kähler settings analogous to Kobayashi–Hitchin correspondences, exploring relations with harmonic bundles on quasi-projective manifolds, and extending the affine/Hessian methods of the curve theory to higher-dimensional affine manifolds and more general non-compact settings (Geng et al., 15 Jul 2025, Collins et al., 2014). A plausible implication is that Hermitian-Poisson metrics supply a flexible replacement for Hermitian-Einstein metrics precisely in settings where projective flatness persists but standard Kähler tools do not.