Griffiths' form-level positivity question: weak positivity of Schur forms for Griffiths-positive vector bundles

Griffiths (1969) asked which polynomials in the Chern classes are positive for Griffiths-positive or ample vector bundles; at the cohomology-class level Fulton and Lazarsfeld proved the answer to be exactly the nonnegative linear combinations of Schur polynomials. The differential-form-level counterpart - whether such combinations, evaluated on the Chern forms of a Griffiths (semi)positive Hermitian metric, are pointwise positive as (p,p)-forms - remains open in general. The source paper (Ping Li, Math. Ann. 2021) proved that for Bott-Chern nonnegative metrics all Schur forms are strongly nonnegative and satisfy the two-sided bounds $0\le_s c_i\le_s c_\lambda\le_s c_1^i$, presenting these results as positive evidence toward Griffiths' form-level question, which its introduction describes as 'still largely unknown'. Later work settled major special cases - signed Segre forms (Guler), rank-three polynomials (Fagioli), Nakano and strongly decomposable positivity (Finski, Wan), and most recently the third Chern form together with the full rank-3/threefold case (Wan 2026) - while confirming that the general question remains unresolved.

Background

Let $(E,h)$ be a Hermitian holomorphic vector bundle of rank $r$ over a complex manifold $X$ of complex dimension $n$. The Chern connection of the metric $h$ has curvature $\Theta_E$, which in a local holomorphic frame is an $r\times r$ matrix $\Omega$ of $(1,1)$-forms. The Chern forms are defined by the Chern-Weil expansion $\det\Big(t\,I_r+\frac{\sqrt{-1}}{2\pi}\,\Omega\Big)=:\sum_{i=0}^{r} c_i(E,h)\,t^{r-i},$ so that $c_i(E,h)$ is a closed real $(i,i)$-form representing the topological Chern class $c_i(E)\in H^{2i}(X,\mathbb{Z})$.

Several nested notions of curvature positivity are relevant. The bundle $(E,h)$ is Griffiths semipositive (resp. Griffiths positive) if $\langle\sqrt{-1}\,\Theta_E(\xi,\bar{\xi})v,v\rangle_h\ge 0$ (resp. $>0$ for $\xi\neq 0$ and $v\neq 0$) for every $(1,0)$-tangent vector $\xi$ and every fiber vector $v$. Stronger conditions are Nakano and dual Nakano (semi)positivity, where the curvature is required to be a (semi)positive Hermitian form on $T^{1,0}X\otimes E$ (resp. on $T^{1,0}X\otimes\bar{E}$). Bott-Chern nonnegativity, introduced in the study of value distribution theory, requires that around every point there exist a unitary frame in which the curvature matrix factors as $\Omega=A\wedge A^{t}$ for some matrix $A$ of $(1,0)$-forms; globally generated bundles always admit such metrics (as quotient metrics of trivial bundles), and Finski proved that Bott-Chern nonnegativity is equivalent to dual Nakano nonnegativity. On the algebraic side, $E$ is ample if the tautological line bundle on its projectivization is ample, and numerically effective (nef) if that line bundle has nonnegative degree on every curve. Griffiths positivity implies ampleness; whether conversely every ample bundle carries a Griffiths positive metric is a different famous conjecture of Griffiths, not the subject of this problem.

For differential forms, fix $1\le p\le n$ and $q=n-p$. A real $(p,p)$-form $u$ is weakly nonnegative if $u\wedge(\sqrt{-1})^{q^2}\beta\wedge\bar{\beta}$ is a nonnegative volume form for every decomposable $(q,0)$-form $\beta=\beta_1\wedge\cdots\wedge\beta_q$, and weakly positive if this volume form is positive for every nonzero decomposable $\beta$; $u$ is Hermitian positive if the same holds for every $\beta\in\Lambda^{q,0}$, which is equivalent to $u$ being expressible as a finite sum $\sum_s(\sqrt{-1})^{p^2}\xi_s\wedge\bar{\xi}_s$ with arbitrary $(p,0)$-forms $\xi_s$; and $u$ is strongly positive if such a representation exists with all $\xi_s$ decomposable. The three cones are nested, coincide for $p\in\{0,1,n-1,n\}$, and are strictly different for $2\le p\le n-2$. Crucially, the product of two weakly nonnegative forms need not be weakly nonnegative (Błocki-Pliś), whereas products of Hermitian positive forms remain Hermitian positive - the structural fact behind both the known positive results and the current obstructions. Terminology varies in the literature: Li's 'nonnegative' ($\ge 0$) and 'strongly nonnegative' ($\ge_s 0$) correspond respectively to weak nonnegativity and Hermitian positivity as defined here.

To a partition $\lambda$ with parts at most $r$ one attaches the Schur polynomial $S_\lambda(c_1,\ldots,c_r):=\det(c_{\lambda_i-i+j})$ of weighted degree $|\lambda|$; the Chern monomials and the (signed) Segre polynomials - the coefficients of the formal inverse of the total Chern polynomial - are special cases. Evaluating $S_\lambda$ on the Chern forms of $(E,h)$ yields the Schur forms $S_\lambda(c(E,h))$.

At the level of cohomology classes the positivity question is completely understood: Fulton and Lazarsfeld proved that a real polynomial is numerically positive for every ample vector bundle exactly when it is a nonnegative linear combination of Schur polynomials, and these inequalities were later extended to nef bundles over compact Kähler manifolds. Griffiths, who introduced Griffiths positivity in 1969, asked the analogous question at the level of differential forms: which polynomials in the Chern forms are pointwise positive for Griffiths-positive or ample bundles? He proved that the second Chern form of a rank-2 Griffiths positive bundle is positive (the general-rank case follows by a reduction recorded by Fagioli), and Guler proved that all signed Segre forms are weakly positive for Griffiths semipositive bundles. Finski reformulated the question equivalently as a statement about positive linear maps between matrix spaces and their mixed discriminants, and proved that Schur forms are positive for Nakano and dual Nakano positive bundles. Diverio-Fagioli and Fagioli obtained weak positivity of further polynomials through pointwise Gysin pushforward formulae on flag bundles. Wan proved the Schur forms are weakly positive under 'strongly decomposable positivity' conditions interpolating between Griffiths and Nakano positivity, and most recently settled the three-dimensional case of Finski's mixed-discriminant problem, deducing weak positivity of the third Chern form for Griffiths positive bundles of arbitrary rank and a complete affirmative answer to Griffiths' question for rank-three bundles over complex threefolds.

Li's 2021 paper, the source paper, proves that for Bott-Chern nonnegative metrics all Schur forms are strongly nonnegative (its Proposition 3.1) and that all monomials $c_\lambda(E,h)=\prod_j c_{\lambda_j}(E,h)$ in the Chern forms obey the two-sided strong-nonnegativity bounds of its Theorem 3.2, with the Chern-number consequences $0\le c_n[E]\le c_\lambda[E]\le c_1^n[E]$ and geometric applications to homogeneous manifolds and submanifolds of complex tori. The paper presents these results as evidence toward Griffiths' form-level question - its introduction describes that question as 'still largely unknown' - and its multiplicative method breaks down for weaker curvature positivity, where products of weakly positive forms may fail to be positive. The open problem identified in the paper is the general form-level question itself; the paper does not propose extending the two-sided bounds beyond the Bott-Chern (equivalently, dual Nakano nonnegative) regime.

Problem Statement

The problem is Griffiths' form-level question (1969), in its standard modern formulation. Let $P\in\mathbb{R}[c_1,\ldots,c_r]$ be a nonzero nonnegative linear combination of Schur polynomials of weighted degree $k$, and let $(E,h)$ be a Griffiths positive Hermitian holomorphic vector bundle of rank $r$ over a complex manifold $X$ with $\dim_{\mathbb{C}}X=n\ge k$. Is the $(k,k)$-form $P\big(c_1(E,h),\ldots,c_r(E,h)\big)$ weakly positive for every such $(E,h)$, and weakly nonnegative for every Griffiths semipositive $(E,h)$ (the same condition with the non-strict curvature inequality)? The complete form asks for a characterization: determine exactly the cone of real polynomials whose evaluations on the Chern forms are weakly positive for every Griffiths positive bundle - both whether the Schur cone is contained in it and whether anything beyond the Schur cone belongs to it. By Finski's theorem this is equivalent to a finite-dimensional statement: for complex vector spaces $V,W$ of equal dimension and a positive linear map $H:\mathrm{End}(V)\to\mathrm{End}(W)$ (one carrying positive semidefinite operators to positive semidefinite operators), the double mixed discriminant $D_W\circ H^{\otimes\dim W}\circ D_V^{*}$ - with $D_V$ the polarization of the determinant and $D_V^{*}$ its dual map - is a positive real number.

A negative answer is witnessed by a specific Griffiths positive bundle, a specific nonnegative combination of Schur polynomials, and a specific point and decomposable vector on which the evaluation is negative. The claim is stated in full generality - all ranks $r$ and all complex dimensions $n\ge k$, with no base-geometry or global-generation hypothesis; the settled special cases (rank three over threefolds, Nakano-type positivity regimes) do not resolve it.

References

1. Griffiths, P. A.: Hermitian differential geometry, Chern classes, and positive vector bundles. In: Global Analysis (Papers in Honor of K. Kodaira), pp. 185-252. Princeton University Press, Princeton, 1969. DOI 10.1515/9781400871230-011, https://doi.org/10.1515/9781400871230-011
2. Bott, R., Chern, S.-S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114 (1965), 71-112. DOI 10.1007/BF02391818, https://doi.org/10.1007/BF02391818
3. Fulton, W., Lazarsfeld, R.: Positive polynomials for ample vector bundles. Ann. of Math. (2) 118 (1983), 35-60. DOI 10.2307/2006953, https://doi.org/10.2307/2006953
4. Guler, D.: On Segre forms of positive vector bundles. Canad. Math. Bull. 55 (2012), 108-113. DOI 10.4153/cmb-2011-100-6, https://doi.org/10.4153/cmb-2011-100-6
5. Błocki, Z., Pliś, S.: Squares of positive (p,p)-forms. C. R. Math. Acad. Sci. Paris 351 (2013), no. 1-2, 27-32. DOI 10.1016/j.crma.2013.01.009, https://doi.org/10.1016/j.crma.2013.01.009
6. Li, P.: Nonnegative Hermitian vector bundles and Chern numbers. Math. Ann. 380 (2021), 21-41. DOI 10.1007/s00208-020-01974-x, https://doi.org/10.1007/s00208-020-01974-x (arXiv:1702.01701)
7. Finski, S.: On characteristic forms of positive vector bundles, mixed discriminants, and pushforward identities. J. Lond. Math. Soc. (2) 106 (2022), 1539-1579. DOI 10.1112/jlms.12605, https://doi.org/10.1112/jlms.12605 (arXiv:2009.13107)
8. Diverio, S., Fagioli, F.: Pointwise universal Gysin formulae and applications towards Griffiths' conjecture. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), 1597-1624. DOI 10.2422/2036-2145.202011_021, https://doi.org/10.2422/2036-2145.202011_021 (arXiv:2009.14587)
9. Fagioli, F.: A note on Griffiths' conjecture about the positivity of Chern-Weil forms. Differential Geom. Appl. 81 (2022), 101848. DOI 10.1016/j.difgeo.2022.101848, https://doi.org/10.1016/j.difgeo.2022.101848 (arXiv:2012.12815)
10. Xiao, J.: On the positivity of high-degree Schur classes of an ample vector bundle. Sci. China Math. 65 (2022), 51-62. DOI 10.1007/s11425-020-1868-7, https://doi.org/10.1007/s11425-020-1868-7 (arXiv:2007.12425)
11. Li, P., Zheng, F.: Chern class inequalities on polarized manifolds and nef vector bundles. Int. Math. Res. Not. 2022 (2022), 6262-6288. DOI 10.1093/imrn/rnaa317, https://doi.org/10.1093/imrn/rnaa317 (arXiv:2004.09224)
12. Wan, X.: Positivity of Schur forms for strongly decomposably positive vector bundles. Forum Math. Sigma 12 (2024), Paper No. e9, 28 pp. DOI 10.1017/fms.2023.125, https://doi.org/10.1017/fms.2023.125 (arXiv:2301.03950)
13. Fagioli, F.: Universal vector bundles, push-forward formulas, and positivity of characteristic forms. Math. Nachr. 298 (2025), no. 12, 3995-4016. DOI 10.1002/mana.70061, https://doi.org/10.1002/mana.70061 (arXiv:2210.11157)
14. Li, P.: Chern numbers on positive vector bundles and combinatorics. Int. Math. Res. Not. 2025 (2025), no. 15, rnaf228. DOI 10.1093/imrn/rnaf228, https://doi.org/10.1093/imrn/rnaf228 (arXiv:2501.08833)
15. Fagioli, F., Mainenti, A.: Some criteria for positive forms and applications. Preprint (2025), arXiv:2502.17317, https://arxiv.org/abs/2502.17317
16. Wan, X.: Positivity of the third Chern form for Griffiths positive vector bundles. Preprint (2026), arXiv:2601.10424, https://arxiv.org/abs/2601.10424
17. Demailly, J.-P.: Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. Sb. Math. 212 (2021), no. 3, 305-318. DOI 10.1070/SM9387, https://doi.org/10.1070/SM9387 (arXiv:2002.02677)
18. Murakami, R.: An analytic proof of Griffiths' conjecture on compact Riemann surfaces. Preprint (2025), arXiv:2509.23201, https://arxiv.org/abs/2509.23201