Griffiths Positivity in Complex Geometry

• Griffiths positivity is defined by a nonnegative Hermitian curvature tensor for holomorphic vector bundles, linking curvature conditions with vanishing theorems and cohomological invariants.
• Recent advances reveal its implications for the positivity of Schur polynomials in Chern forms and establish universal pushforward formulas on flag bundles.
• The concept underpins practical applications such as vanishing theorems and L² extension results, with analytic and metric characterizations enhancing its role in complex differential geometry.

Griffiths positivity is a fundamental notion of Hermitian curvature positivity for holomorphic vector bundles over complex manifolds. It occupies a central role in complex differential geometry, Hodge theory, and algebraic geometry, serving as a bridge between curvature conditions and positivity properties of characteristic forms, cohomological invariants, and vanishing theorems. Originating in the work of Phillip Griffiths on ampleness and vector bundle positivity, the concept has evolved to incorporate operator-theoretic, analytic, and pluripotential perspectives, with recent advances elucidating its implications for Schur polynomials in Chern forms, flag bundles, and the behavior of direct image sheaves.

1. Foundational Definition and Basic Properties

Let $(E, h) \to X$ be a holomorphic vector bundle of rank $r$ over an $n$-dimensional complex manifold, equipped with a smooth Hermitian metric $h$. The Chern connection has curvature tensor

$$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$

with local representation in a frame $\{e_\alpha\}$. $(E,h)$ is Griffiths semipositive (resp., positive) if, for every $x \in X$, every nonzero tangent vector $\tau \in T_{X,x}$, and every nonzero vector $v\in E_x$,

$r$0

(resp., $r$1). This defines a Hermitian form on $r$2, strictly intermediate between Nakano and dual Nakano positivity, and generalizes both line bundle positivity and the classical notion of Kähler metrics (Fagioli, 2020).

Analytically, for any local holomorphic section $r$3 of $r$4, the function $r$5 is plurisubharmonic if and only if $r$6 is Griffiths-negative, i.e., the dual metric $r$7 is Griffiths-positive (Varolin, 3 Aug 2025, Deng et al., 2018).

2. Positivity of Characteristic Forms and Schur Polynomials

The central conjecture due to Griffiths posits that all Schur polynomials in the Chern forms of a Griffiths positive bundle are positive as $r$8-forms ("Schur-positivity"). Let $r$9 denote the $n$0-th Chern form, and $n$1 the Schur polynomial corresponding to the partition $n$2. For example, in rank 3,

$n$3

is the Schur form associated to the partition $n$4.

Substantial progress has been made for low ranks:

• Rank 2: Griffiths showed that $n$5 is positive.
• Rank 3, Dimension 3: All Schur forms are proven weakly positive for Griffiths positive bundles (Wan, 15 Jan 2026). The key tool is the analysis of "double mixed discriminants" associated to positive linear maps between spaces of $n$6 matrices, using operator-scaling and combinatorial identities. The criterion reduces the positivity of Schur forms to positivity properties of mixed discriminants.

A general chain of inequalities among Chern forms holds in rank 3 (Fagioli, 2020): $n$7 pointwise as $n$8-forms.

Universal push-forward (Gysin) formulas, originally developed for cohomology classes, are shown to hold pointwise for Chern–Weil forms as well, allowing the construction of large cones of positive polynomials in the Chern forms under Griffiths semipositivity (Diverio et al., 2020, Fagioli, 2022). More explicit positive representatives are constructed for Schur classes of bidegree $n$9 when $h$0 is ample (Xiao, 2020).

3. Operator-Theoretic and Pushforward Perspectives

Operator theory provides a natural language for curvature positivity. The curvature tensor can be encoded as a positive map $h$1, with positivity meaning that $h$2 sends positive semidefinite matrices to positive definite ones. The mixed and double mixed discriminants of these operators play a pivotal role: Schur-form positivity on a Griffiths positive bundle is equivalent to the positivity of such discriminants (Finski, 2020, Wan, 15 Jan 2026).

A universal pushforward formula for flag bundles establishes that for any polynomial in the Chern forms of universal bundles over flag bundles, fiber integration yields a universal polynomial in the Chern forms of $h$3 holding pointwise (not just in cohomology) (Diverio et al., 2020, Fagioli, 2022). This provides a robust mechanism for transferring positivity from bundles on flag varieties to characteristic forms on the base, supporting the Griffiths conjecture for broad subfamilies of polynomials ("Gysin subcone") (Diverio et al., 2020).

4. Analytic, Pluripotential, and Metric Characterizations

Analytic characterizations relate Griffiths positivity to plurisubharmonicity: the dual metric $h$4 is Griffiths positive if and only if, for every local holomorphic section $h$5 of $h$6, $h$7 is plurisubharmonic (Deng et al., 2018, Watanabe, 2024).

In the context of singular Hermitian metrics, a metric $h$8 is said to be Griffiths semi-positive if for every local holomorphic section $h$9 of $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$0, $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$1 is plurisubharmonic in the sense of distributions. This is equivalent to approximating $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$2 by a decreasing sequence of smooth Griffiths semi-negative metrics. Further, $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$3-trace positivity, defined via contraction of the curvature with an auxiliary Hermitian metric $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$4, is shown to be equivalent to Griffiths positivity (for all $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$5) (Watanabe, 2024).

Practical consequences include vanishing theorems: any Griffiths quasi-positive metric on $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$6 (even singular) leads to strict maximality of $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$7 and vanishing of all $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$8-th cohomology of tensor and wedge powers of $$\Theta(E,h) = \left(\Theta_{\alpha\bar\beta}^{\ \,\,\gamma\bar\delta}\right) \in \mathcal{A}^{1,1}(X, \mathrm{End}(E)),$$9 (Watanabe, 2024).

5. Geometric Constructions, Representation-Theoretic Aspects, and Homogeneous Models

On flag manifolds and their associated bundles, explicit classification results identify all Kähler flag manifolds with Griffiths semi-positive curvature as precisely the Hermitian symmetric spaces, equipped with canonical invariant metrics (Galindo et al., 27 Jun 2025). The Chern curvature is expressed explicitly in terms of root-theoretic data and weights on the isotropy representation, with underpinnings in the representation theory of Lie groups. No genuinely quasi-Kähler flag manifolds admit Griffiths semi-positive metrics except for products of Hermitian symmetric spaces.

Sufficient criteria for Griffiths positivity employ direct image and $\{e_\alpha\}$0-metric constructions. For instance, a positive Hermitian metric on $\{e_\alpha\}$1, whose fiberwise Kähler–Einstein condition holds, induces a Griffiths positive $\{e_\alpha\}$2-metric on $\{e_\alpha\}$3; the relative Kähler–Ricci flow is proposed as a mechanism to construct such metrics in general (Naumann, 2017).

6. Hermitian–Yang–Mills Approach and Elliptic PDE Systems

Demailly introduces a Hermitian–Yang–Mills–Monge–Ampère system for Hermitian metrics on $\{e_\alpha\}$4, combining determinant constraints on the curvature operator and Hermite–Einstein-type trace conditions. Solutions would produce dual Nakano positive (and hence Griffiths positive) metrics if they exist for all $\{e_\alpha\}$5. The main challenge remains the extension to $\{e_\alpha\}$6, which would affirm the implication: ampleness $\{e_\alpha\}$7 Griffiths positivity (Demailly, 2020).

7. Applications, Extension Theorems, and Open Problems

Griffiths positivity underpins vanishing theorems and $\{e_\alpha\}$8-extension results for holomorphic sections. However, it is not sufficient for the Ohsawa–Takegoshi $\{e_\alpha\}$9 extension of top-degree forms, for which Nakano positivity is necessary; counterexamples are given by universal quotient bundles on projective spaces (Varolin, 3 Aug 2025).

Furthermore, the positivity of direct image bundles ($(E,h)$0) under Berndtsson's theorem is known to imply (Nakano, hence Griffiths) semi-positivity, and recent quantitative converses establish that Griffiths semi-positivity for all twisted direct images forces semi-positivity of $(E,h)$1 (Xu et al., 19 Jan 2026).

In the context of Bismut connections on non-Kähler manifolds, a "Bismut–Griffiths positivity" is developed, but its preservation under Hermitian curvature flows exhibits nuanced behavior, being flow-dependent and less robust than its Kähler counterpart (Barbaro, 2021).

Persistent open problems include:

• General (rank $(E,h)$2) affirmative solution of the Griffiths conjecture for Schur form positivity;
• Loosened criteria for characterizing Griffiths positivity purely by positivity of characteristic forms, which is known to fail in higher rank (Finski, 2020);
• The existence of Griffiths positive metrics on all ample bundles;
• The behavior of positivity notions under degenerations and singular metrics.

Summary Table: Key Results

Notion/Setting	Main Result	Paper(s)
Griffiths positivity (definition)	$(E,h)$3	(Fagioli, 2020, Varolin, 3 Aug 2025)
Positivity of Schur forms (rank 3)	All Schur forms weakly positive in rank 3, $(E,h)$4	(Wan, 15 Jan 2026, Fagioli, 2020)
Pointwise Gysin formula	Universal pushforward holds for Chern–Weil forms	(Diverio et al., 2020, Fagioli, 2022)
Analytic characterization (psh)	$(E,h)$5 psh $(E,h)$6 $(E,h)$7 Griffiths positive	(Deng et al., 2018, Watanabe, 2024)
Hermitian–Yang–Mills PDE system	Existence $(E,h)$8 dual Nakano positivity	(Demailly, 2020)
Extension theorems: necessity	Griffiths positivity insufficient for $(E,h)$9 extension	(Varolin, 3 Aug 2025)
Invariant metrics on flag manifolds	Only Hermitian symmetric spaces with unique invariant metric	(Galindo et al., 27 Jun 2025)

• (Fagioli, 2020) "A note on Griffiths' conjecture about the positivity of Chern-Weil forms"
• (Diverio et al., 2020) "Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture"
• (Fagioli, 2022) "Universal vector bundles, push-forward formulae and positivity of characteristic forms"
• (Wan, 15 Jan 2026) "Positivity of Schur forms for Griffiths positive vector bundles of rank three over complex threefolds"
• (Finski, 2020) "On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities"
• (Watanabe, 2024) "$x \in X$0-trace and Griffiths positivity for singular Hermitian metrics"
• (Deng et al., 2018) "New characterizations of plurisubharmonic functions and positivity of direct image sheaves"
• (Varolin, 3 Aug 2025) "Positivity and $x \in X$1 Extension"
• (Galindo et al., 27 Jun 2025) "Curvature positivity for Kähler and quasi-Kähler flag manifolds"
• (Naumann, 2017) "An approach to Griffiths conjecture"
• (Demailly, 2020) "Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles"
• (Barbaro, 2021) "Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows"
• (Xu et al., 19 Jan 2026) "A converse of Berndtsson's theorem on the positivity of direct images"
• (Xiao, 2020) "On the positivity of high-degree Schur classes of an ample vector bundle"
• (Chen, 2022) "Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries"