Griffiths' 1969 question on the positivity of Schur forms for Griffiths-positive vector bundles and Finski's equivalent mixed-discriminant problem for positive linear maps

In 1969 Griffiths asked whether the characteristic forms of a Hermitian holomorphic vector bundle with Griffiths-positive curvature are themselves positive differential forms; the precise conjecture is that every nonnegative linear combination of Schur forms of a Griffiths-positive bundle is weakly positive. Finski (2022) proved this question equivalent to a finite-dimensional problem of operator theory: for a positive linear map $H:\mathrm{End}(V)\to\mathrm{End}(W)$ with $\dim V=\dim W$, is the double mixed discriminant $\mathrm{D}_W\circ H^{\otimes\dim W}\circ\mathrm{D}_V^*(1)$ a positive real number? Wan's 2026 preprint settles the case $\dim V=\dim W=3$ — and hence Griffiths' question for bundles of rank three over complex threefolds — together with weak positivity of the third Chern form for bundles of arbitrary rank $r\ge 3$. The general question remains open as of August 2026, with established open instances in rank and degree four.

Background

Let $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ over a complex manifold $X$ of complex dimension $n$. The Chern connection of $h^E$ has curvature $R^E\in A^{1,1}(X,\mathrm{End}(E))$, and the Chern forms $c_k(E,h^E)\in A^{k,k}(X)$ are defined as the coefficients of $\det(\mathrm{Id}_E+\frac{\sqrt{-1}}{2\pi}R^E)$; they are closed real $(k,k)$-forms representing the Chern classes of $E$. The bundle is called Griffiths positive if, for every point $z\in X$, every nonzero tangent vector $\xi\in T_z^{1,0}X$, and every nonzero vector $v\in E_z$, the contraction $R^E_{i\bar j\alpha\bar\beta}v^i\bar v^j\xi^\alpha\bar\xi^\beta$ is a positive real number. This is the weakest of the standard curvature positivity notions for vector bundles: it is implied by Nakano and dual Nakano positivity — stronger conditions requiring the curvature to be positive as a Hermitian form on all tensors of $E\otimes T^{1,0}X$ and of $E^{*}\otimes T^{1,0}X$ respectively, not only on decomposable ones — by Bott–Chern positivity (a companion curvature condition which Finski showed to be equivalent, in its nonnegative form, to dual Nakano nonnegativity), and by ampleness-type conditions (positivity in the algebro-geometric sense of having sufficiently many global sections), but implies much less.

Positivity of differential forms is itself a subtle notion. A real $(k,k)$-form $u$ on $X$ is called weakly positive if for every nonzero decomposable $(n-k,0)$-form $\beta=\beta_1\wedge\dots\wedge\beta_{n-k}$, the top-degree form $u\wedge(\sqrt{-1})^{(n-k)^2}\beta\wedge\bar\beta$ is a positive volume form; weak positivity is the weakest of the classical positivity notions for forms (weak, positive, strong), which coincide only in degrees $0,1,n-1,n$.

The characteristic forms adapted to this question are the Schur forms. For a partition $\lambda=(\lambda_1,\dots,\lambda_r)$ with $r\ge\lambda_1\ge\dots\ge\lambda_r\ge0$ and $|\lambda|=k$, the Schur form is the determinant $P_\lambda(c(E,h^E))=\det(c_{\lambda_i-i+j}(E,h^E))_{1\le i,j\le r}$, a closed real $(k,k)$-form generalizing the Chern forms (the partitions $(k)$) and the signed Segre forms (up to sign, the coefficients of the inverse of the total Chern form, i.e. the partitions $(1,\dots,1)$). On the purely topological side, Fulton and Lazarsfeld proved in 1983 that all Schur polynomials are numerically positive for ample vector bundles; Griffiths proposed in 1969 the analytic counterpart for Griffiths positivity, asking whether the corresponding differential forms are weakly positive.

Several cases are known. Griffiths himself proved that the second Chern form $c_2(E,h^E)$ is positive for Griffiths-positive bundles. Güler (2012) established weak positivity of all signed Segre forms. Diverio and Fagioli (2022), via pointwise universal Gysin (pushforward) formulae on flag bundles, obtained weak positivity for further polynomials in the Chern forms, and Fagioli (2022) proved, in rank three, weak nonnegativity of $P_{(2,1,0)}=c_1c_2-c_3$ for Griffiths semipositive bundles together with the inequalities $c_1^3\ge c_1c_2\ge c_3$. Under stronger curvature assumptions everything is known: Finski (2022) proved nonnegativity of all Schur forms for Nakano-nonnegative bundles and positivity for (dual) Nakano-positive ones, and Wan (2024) handled bundles whose curvature is strongly decomposably positive, a condition between Griffiths and Nakano positivity.

The decisive structural insight is due to Finski (2022). The mixed discriminant $\mathrm{D}_V:\mathrm{End}(V)^{\otimes r}\to\mathbb{C}$ is the polarization of the determinant — the symmetric $r$-linear form on $r\times r$ matrices satisfying $\mathrm{D}_V(A,\dots,A)=\det A$ — and $\mathrm{D}_V^*$ denotes its dual map. A linear map $H:\mathrm{End}(V)\to\mathrm{End}(W)$ is positive if it sends every nonzero positive semidefinite endomorphism of $V$ to a positive definite endomorphism of $W$. Writing $H(E_{i\bar j})=B_{i\bar j}\in\mathrm{End}(W)$ for the images of the matrix units, positivity of $H$ is equivalent to the Griffiths-type pencil condition $\sum_{i,j}B_{i\bar j}\xi^i\bar\xi^j\succ0$ for every $0\ne\xi\in\mathbb{C}^r$. Finski proved that Griffiths' question, in its aggregate form over all ranks, is equivalent to the statement that $\mathrm{D}_W\circ H^{\otimes\dim W}\circ\mathrm{D}_V^*(1)$ is positive for every positive linear map $H$ with $\dim V=\dim W$; he also proved the case $\dim V=\dim W=2$. The case $\dim V=\dim W=3$ — the first dimension in which positive maps need not be decomposable (i.e. expressible as a sum of maps of the form $X\mapsto AXB^{*}$ and $X\mapsto AX^{\mathrm{t}}B^{*}$ with fixed matrices $A,B$), as shown by Choi's 1975 example — was settled affirmatively by Wan (2026), who combined a doubly stochastic normalization of $H$ via Gurvits-type operator scaling (an iterative rescaling of the domain and codomain that brings a positive map into doubly stochastic normal form while preserving positivity) with spherical moment identities and Schur's inequality on the eigenvalues of the Hermitian matrix $C(\xi)=(\xi^*B_{i\bar j}\xi)$. This yields weak positivity of all Schur forms for Griffiths-positive bundles of rank three over complex threefolds, and, by a principal-minor argument, weak positivity of the third Chern form for Griffiths-positive bundles of arbitrary rank $r\ge3$. For rank four the same averaging strategy is obstructed: a remainder built from traces along 4-cycles is not expressible as a symmetric polynomial of $C(\xi)$, so the general question — established open instances of which lie in rank and degree four — remains open.

Problem Statement

Problem (Griffiths, 1969; equivalent finite-dimensional form due to Finski, 2022). Let $H:\mathrm{End}(V)\to\mathrm{End}(W)$ be a positive linear map between the spaces of endomorphisms of complex vector spaces $V,W$ with $\dim V=\dim W=r$, i.e. $H$ sends every nonzero positive semidefinite endomorphism of $V$ to a positive definite endomorphism of $W$. Let $\mathrm{D}_V$ be the mixed discriminant (the polarization of the determinant) and $\mathrm{D}_V^*$ its dual map. Is the real number

$\mathrm{D}_W\circ H^{\otimes r}\circ \mathrm{D}_V^*(1)$

positive for every such $H$ and every $r$? By Finski's equivalence theorem, an affirmative answer in all ranks holds if and only if the following geometric statement holds: for every Griffiths-positive Hermitian holomorphic vector bundle $(E,h^E)$ over a complex manifold $X$ of dimension $n$, and every nonzero polynomial $P$ that is a nonnegative linear combination of Schur polynomials of weighted degree $k\le n$ in the Chern classes, the differential form $P(c_1(E,h^E),\dots,c_r(E,h^E))$ is weakly positive on $X$. The question is whether this positivity holds in full generality, with no restriction on rank, dimension, or degree, or whether some rank $r$ admits an explicit counterexample. The already-known cases are rank three over complex threefolds and the third Chern form in all ranks (Wan 2026), all signed Segre forms (Güler 2012), and the second Chern form (Griffiths 1969); the general question remains open, with established open instances in rank and degree four — for instance the weak positivity of $c_4$ for Griffiths-positive bundles of rank $\ge4$.

References

1. P. A. Griffiths, 'Hermitian differential geometry, Chern classes, and positive vector bundles', in Global Analysis (Papers in Honor of K. Kodaira), Princeton University Press, 1969, pp. 185–251. DOI: 10.1515/9781400871230-011, https://doi.org/10.1515/9781400871230-011
2. R. Bott and S. S. Chern, 'Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections', Acta Mathematica 114 (1965) 71–112. DOI: 10.1007/bf02391818, https://doi.org/10.1007/bf02391818
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4. Dincer Guler, 'On Segre Forms of Positive Vector Bundles', Canadian Mathematical Bulletin 55 (2012), no. 1, 108–113. DOI: 10.4153/cmb-2011-100-6, https://doi.org/10.4153/cmb-2011-100-6
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7. Simone Diverio and Filippo Fagioli, 'Pointwise universal Gysin formulae and applications towards Griffiths' conjecture', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze XXIII (2022) 1597–1624. DOI: 10.2422/2036-2145.202011_021, arXiv:2009.14587, https://arxiv.org/abs/2009.14587
8. Xueyuan Wan, 'Positivity of Schur forms for strongly decomposably positive vector bundles', Forum of Mathematics, Sigma 12 (2024) e9. DOI: 10.1017/fms.2023.125, arXiv:2301.03950, https://arxiv.org/abs/2301.03950
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10. Xueyuan Wan, 'Positivity of the third Chern form for Griffiths positive vector bundles', arXiv:2601.10424 (January 2026). DOI: 10.48550/arXiv.2601.10424, https://arxiv.org/abs/2601.10424
11. Filippo Fagioli and Asia Mainenti, 'Some criteria for positive forms and applications', arXiv:2502.17317 (2025), https://arxiv.org/abs/2502.17317
12. Ping Li, 'Chern numbers on positive vector bundles and combinatorics', International Mathematics Research Notices (2025). DOI: 10.1093/imrn/rnaf228, arXiv:2501.08833, https://doi.org/10.1093/imrn/rnaf228, https://arxiv.org/abs/2501.08833