Griffiths' conjecture on the positivity of Chern–Weil (Schur) forms for Griffiths semipositive vector bundles

Let $(E,h)$ be a Hermitian holomorphic vector bundle whose curvature is Griffiths positive. Griffiths conjectured in 1969 that every positive linear combination of Schur polynomials evaluated on the Chern forms of $(E,h)$ — the Schur cone — consists of Hermitian positive differential forms (the recent literature commonly treats the corresponding statement for Griffiths semipositive bundles). Despite the conjecture being verified for globally generated and (dual) Nakano positive bundles, for a push-forward sub-cone, and most recently for all Schur forms of rank-3 bundles over complex threefolds, the conjecture remains open in general: no proof and no counterexample is known for arbitrary rank, and even the correct pointwise positivity notion for the full cone is unsettled as a related question. The problem is to settle Griffiths' conjecture in full generality.

Background

A Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $X$ carries a unique Chern connection, whose curvature $\Theta(E,h)$ is a matrix of $(1,1)$-forms with values in the endomorphisms of $E$. The bundle is Griffiths (semi)positive when the Hermitian form $(v,\Theta(E,h)(w\otimes u))$ is (semi)positive definite on each fiber; stronger curvature conditions include Nakano and dual Nakano positivity, which control the full tensor $\Theta$ rather than only its restriction to decomposable vectors. Via Chern–Weil theory, invariant symmetric polynomials in the curvature produce the Chern forms $c_j(E,h)$, and more generally the Schur forms $S_\sigma(E,h)$ obtained by evaluating a Schur polynomial $S_\sigma$ (the character of an irreducible representation of $GL_r$, indexed by a partition $\sigma$) on the Chern forms. The cone of positive linear combinations of Schur forms is called the Schur cone; by the classical theory it is exactly the cone of 'positive polynomials' in the Chern forms.

For $(p,p)$-forms on a complex vector space several inequivalent positivity notions exist, and they are ordered by strength: a form is weakly positive (often just 'positive') if it pairs nonnegatively with all decomposable positive $(n-p,n-p)$-forms; Hermitian positive if it is a nonnegative sum of forms $\alpha\wedge\bar\alpha$ with $\alpha$ of type $(p,0)$; and strongly positive if it is a nonnegative combination of decomposable positive forms. Strong positivity implies Hermitian positivity implies weak positivity, and both inclusions are strict (in middle bidegrees $2\le p\le n-2$; all three notions coincide for $p=0,1,n-1,n$).

Griffiths, in his 1969 paper 'Hermitian differential geometry, Chern classes, and positive vector bundles', proved the positivity of Chern forms for globally generated bundles and then conjectured (p. 247) that for a Griffiths positive bundle the entire Schur cone consists of Hermitian positive forms. This pointwise, form-level statement is strictly stronger than the algebraic counterpart settled by Fulton and Lazarsfeld, who showed that Schur-positive polynomials are numerically positive on ample vector bundles (positivity of integrals over subvarieties, not of the differential forms themselves). Known cases of the form-level conjecture include: $c_2$ for Griffiths positive rank-2 bundles (Griffiths); the signed Segre forms — up to a sign, the Schur forms of the dual bundle $E^*$ (Guler); the full Schur cone for (dual) Nakano (semi)positive bundles (Finski); a sub-cone of push-forward forms from flag bundles, which are strongly positive for Griffiths semipositive bundles (Diverio–Fagioli); and $S_{(2,1,0)}=c_1c_2-c_3$ for Griffiths semipositive rank-3 bundles (Fagioli). Finski proved that an open algebraic problem on positive semidefinite linear preservers between spaces of Hermitian matrices is equivalent to the weak-positivity formulation of Griffiths' question (whether the Schur forms of Griffiths positive bundles are weakly positive); this equivalence does not cover the stronger Hermitian-positivity target. A technical obstruction to current methods is that powers of positive $(2,2)$-forms are not known to be positive, which blocks extending the push-forward technique beyond the known sub-cone.

Problem Statement

Let $(E,h)$ be a Hermitian holomorphic vector bundle of arbitrary rank over a complex manifold $X$, and let $\Theta(E,h)$ be the curvature of its Chern connection. Recall that $(E,h)$ is Griffiths semipositive (resp. positive) if $\langle\Theta(E,h)(v\otimes\xi),v\otimes\xi\rangle_h\ge 0$ (resp. $>0$) for every vector $v\in E_x$ and every $(1,0)$-tangent vector $\xi\in T^{1,0}_xX$, at every point $x\in X$; and that a real $(p,p)$-form is Hermitian positive if it is a nonnegative real linear combination of forms $\alpha\wedge\bar\alpha$ with $\alpha$ of type $(p,0)$. Let $S_\sigma(E,h)$ denote the Schur forms, obtained by evaluating each Schur polynomial $S_\sigma$ (indexed by a partition $\sigma$) on the Chern forms $c_j(E,h)$ of $(E,h)$. Griffiths conjectured (1969, p. 247) that for a Griffiths positive bundle the Schur cone — the cone of differential forms generated by all $S_\sigma(E,h)$ — consists of Hermitian positive $(p,p)$-forms; recent work commonly states the corresponding variant for Griffiths semipositive bundles, and several partial results are proved in that slightly more general setting. Determine whether this conjecture holds in full generality: either prove that every element of the Schur cone of every Griffiths positive Hermitian holomorphic vector bundle is Hermitian positive, or exhibit a Griffiths positive Hermitian holomorphic vector bundle and an element of its Schur cone that fails to be Hermitian positive.

References

1. P. A. Griffiths, 'Hermitian differential geometry, Chern classes, and positive vector bundles', in Global Analysis (Papers in Honor of K. Kodaira), University of Tokyo Press, 1969, pp. 185–251. zbMATH Zbl 0201.24001; DOI 10.1515/9781400871230-011, https://doi.org/10.1515/9781400871230-011
2. Filippo Fagioli, 'A note on Griffiths' conjecture about the positivity of Chern-Weil forms', Differential Geometry and its Applications 81 (2022), 101848, DOI 10.1016/j.difgeo.2022.101848, https://doi.org/10.1016/j.difgeo.2022.101848 (preprint arXiv:2012.12815, https://arxiv.org/abs/2012.12815)
3. Siarhei Finski, 'On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities', Journal of the London Mathematical Society 106 (2022), 1539–1579, DOI 10.1112/jlms.12605, https://doi.org/10.1112/jlms.12605 (preprint arXiv:2009.13107, https://arxiv.org/abs/2009.13107)
4. Simone Diverio and Filippo Fagioli, 'Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture', arXiv:2009.14587, https://arxiv.org/abs/2009.14587 (DOI 10.2422/2036-2145.202011_021)
5. Filippo Fagioli, 'Universal vector bundles, push-forward formulas, and positivity of characteristic forms', Mathematische Nachrichten 298 (2025), 3995–4016, DOI 10.1002/mana.70061, https://doi.org/10.1002/mana.70061 (preprint arXiv:2210.11157, https://arxiv.org/abs/2210.11157)
6. Xueyuan Wan, 'Positivity of Schur forms for strongly decomposably positive vector bundles', arXiv:2301.03950, https://arxiv.org/abs/2301.03950 (DOI 10.1017/fms.2023.125)
7. Xueyuan Wan, 'Positivity of the third Chern form for Griffiths positive vector bundles', arXiv:2601.10424, https://arxiv.org/abs/2601.10424
8. Ping Li, 'Chern numbers on positive vector bundles and combinatorics', arXiv:2501.08833, https://arxiv.org/abs/2501.08833 (DOI 10.1093/imrn/rnaf228)
9. Filippo Fagioli and Asia Mainenti, 'Some criteria for positive forms and applications', arXiv:2502.17317, https://arxiv.org/abs/2502.17317
10. Jian Xiao, 'On the positivity of high-degree Schur classes of an ample vector bundle', Science China Mathematics, DOI 10.1007/s11425-020-1868-7, https://doi.org/10.1007/s11425-020-1868-7
11. Vamsi Pritham Pingali, 'Representability of Chern–Weil forms', Mathematische Zeitschrift 288 (2018), 629–641, DOI 10.1007/s00209-017-1903-2, https://doi.org/10.1007/s00209-017-1903-2
12. William Fulton and Robert Lazarsfeld, 'Positive Polynomials for Ample Vector Bundles', Annals of Mathematics (2) 118 (1983), 35–60, DOI 10.2307/2006953, https://doi.org/10.2307/2006953
13. Zbigniew Błocki and Szymon Pliś, 'Squares of positive (p,p)-forms', Comptes Rendus Mathématique. Académie des Sciences. Paris 351 (2013), no. 1-2, 27–32, MR 3019757, DOI 10.1016/j.crma.2013.01.009, https://doi.org/10.1016/j.crma.2013.01.009