Finski's open problem on the positivity of the double mixed discriminant of a positive linear map, equivalently Griffiths' 1969 question on weak positivity of Schur forms for Griffiths positive vector bundles

Finski (2022) proved that Griffiths' 1969 question on the positivity of characteristic forms - whether every nonzero nonnegative linear combination of Schur polynomials, evaluated in the Chern forms of a Griffiths positive Hermitian holomorphic vector bundle, is weakly positive as a differential form - is equivalent to an operator-theoretic problem on matrix spaces: for every positive linear map $H:\mathrm{End}(V)\to\mathrm{End}(W)$ between endomorphism spaces of equidimensional complex vector spaces, is the double mixed discriminant $D_W\circ H^{\otimes r}\circ D_V^*$ a positive real number? The problem is settled affirmatively for dimensions $r\le 2$ (Finski 2022) and $r=3$ (Wan 2026); consequences include weak positivity of all Schur forms for rank-3 Griffiths positive bundles over complex threefolds and of the third Chern form in arbitrary rank. The case $r\ge 4$ - equivalently Griffiths' question in its remaining full generality - remains open.

Background

Let $(E,h)$ be a Hermitian holomorphic vector bundle of rank $r$ over a complex manifold $X$ of dimension $n$. The Chern curvature $R_E$ is an $\mathrm{End}(E)$-valued $(1,1)$-form, and Chern-Weil theory defines the Chern forms $c_i(E,h)$ through $\det(\mathrm{Id}_E+\frac{\sqrt{-1}}{2\pi}R_E)=\sum_i c_i(E,h)$: each $c_i$ is the $i$-th elementary symmetric function of the curvature matrix entries and is a closed real $(i,i)$-form. The bundle is Griffiths positive (resp. semipositive) if for every point $x\in X$, every nonzero tangent vector $\xi\in T^{1,0}_xX$, and every nonzero $v\in E_x$, one has $h(R_E(\xi,\bar\xi)v,v)>0$ (resp. $\ge 0$); equivalently, writing the curvature as a matrix $\Theta=(\Theta_{i\bar j})$ of $(1,1)$-forms in a local holomorphic frame, the matrix $\sum_{i,j}\Theta_{i\bar j}\xi^i\bar\xi^j$ is positive (semi)definite for every $\xi\ne 0$. Griffiths positivity is the weakest standard curvature positivity notion; Nakano positivity, which requires the curvature to be positive as a Hermitian form on all of $E\otimes T^{*}X$ rather than on decomposable tensors, is strictly stronger.

A real $(k,k)$-form $u$ on $X$ is called weakly positive (resp. weakly nonnegative) if its restriction to every $k$-dimensional complex tangent plane is a positive (resp. nonnegative) volume form for the canonical orientation; this is the pointwise notion of positivity adapted to characteristic forms, and weak positivity can be tested by restriction to $k$-dimensional subspaces.

For a partition $\lambda=(\lambda_1\ge\cdots\ge\lambda_r\ge 0)$ of $k$ with $\lambda_1\le r$, the Schur polynomial $s_\lambda$ is the character of the irreducible representation of $\mathrm{GL}_r$ of highest weight $\lambda$, a symmetric polynomial in $r$ variables computable by the Jacobi-Trudi determinant; evaluated on the Chern forms it produces the Schur form $P_\lambda(c(E,h))$, a closed real $(k,k)$-form generalizing both Chern forms and signed Segre forms. Finski and Wan index these forms by the conjugate partition, so in their notation $P_{(k,0,\ldots,0)}=c_k$ and, in rank 3, $P_{(2,1,0)}=c_1c_2-c_3$ and $P_{(1,1,1)}=c_1^3-2c_1c_2+c_3$.

Two classical results anchor the question. At the cohomological level, Fulton and Lazarsfeld proved that every nonnegative linear combination of Schur polynomials is numerically positive on ample vector bundles. Griffiths then asked for the pointwise differential-geometric refinement: for bundles carrying a Griffiths positive metric, the corresponding characteristic forms should be weakly positive. Griffiths verified $c_2$ by explicit computation; Bott and Chern (1965), extended by Li to all Schur forms, settled the stronger (dual) Nakano nonnegativity condition, which Finski showed to be equivalent to Bott-Chern nonnegativity; Guler proved the question for all signed Segre forms.

Finski's central insight is that Griffiths' question is equivalent to a problem in the operator theory of matrix spaces. For a complex vector space $V$ with $\dim V=r$, the mixed discriminant $D_V:\mathrm{End}(V)^{\otimes r}\to\mathbb{C}$ (often called the Alexandroff mixed discriminant) is the symmetric multilinear polarization of the determinant; in coordinates $D_V(A_1,\ldots,A_r)=\frac{1}{r!}\sum_{\sigma\in S_r}\det(a^{\sigma(i)}_{ik})$. A linear map $H:\mathrm{End}(V)\to\mathrm{End}(W)$ is called non-negative (resp. positive) if it sends every nonzero positive semidefinite endomorphism of $V$ to a positive semidefinite (resp. positive definite) endomorphism of $W$ - in operator-theoretic language, a positive map between matrix algebras. Positive semidefiniteness of the Choi matrix $\sum_{i,j}E_{ij}\otimes H(E_{ij})$ characterizes the strictly stronger notion of complete positivity, not the (non-)negativity defined here, so the class of maps under consideration is genuinely larger than the completely positive one. The curvature of a Hermitian bundle induces at each point a linear operator $H^E_x:\mathrm{End}(T^{1,0}_xX)\to\mathrm{End}(E_x)$, and Finski proved that $(E,h)$ is Griffiths (semi)positive at $x$ if and only if $H^E_x$ is a (non-)negative operator, while conversely every positive operator is locally realized as a curvature operator of some metric. In coordinates, if $B_{i\bar j}=H(E_{i\bar j})$ for the standard matrix units, positivity of $H$ is exactly the Griffiths-type condition $\sum_{i,j}B_{i\bar j}\xi^i\bar\xi^j\succ 0$ for all $\xi\ne 0$, and the double mixed discriminant

$\Phi(H)=D_W\circ H^{\otimes r}\circ D_V^*\in\mathbb{R}$

(where $D_V^*$ is the dual of $D_V$ under $\mathrm{End}(V)\simeq\mathrm{End}(V)^*$) equals $\sum_{\sigma\in S_r}\mathrm{sgn}(\sigma)\,D(B_{1\sigma(1)},\ldots,B_{r\sigma(r)})$. Finski posed the positivity of $\Phi(H)$ as an open problem and proved that its answer coincides with the answer to Griffiths' question.

The state of the art: Finski settled $\dim V=\dim W=2$ and maps satisfying decomposable-type structural conditions. Here a map is called decomposable if it is a finite sum of maps of the form $X\mapsto AXB$ with $A,B$ positive semidefinite, and entrywise if it vanishes on all off-diagonal matrix units; for entrywise maps the statement reduces to the classical mixed-discriminant inequality for positive semidefinite matrices. Diverio-Fagioli and Fagioli obtained further families of positive polynomials via pointwise Gysin pushforward formulae (identities computing the pushforward of polynomials in the Chern forms of tautological bundles along flag-bundle projections at the level of differential forms) and, in rank 3, weak nonnegativity of $P_{(2,1,0)}$ together with the inequalities $c_1^3\ge c_1c_2\ge c_3$; Wan settled strongly decomposably positive bundles; and in January 2026 Wan settled the first genuinely new dimension $\dim V=\dim W=3$ - the first dimension in which positive maps need not be decomposable, by Choi's example - by combining Gurvits' operator scaling (iterated rescalings of domain and codomain bringing $H$ to a doubly stochastic normalization, under which the sign of $\Phi$ is invariant) with a spherical moment identity that reduces $\Phi$ to a sharp Schur inequality for eigenvalues. This yields a complete affirmative answer to Griffiths' question for Griffiths positive bundles of rank 3 over complex threefolds and weak positivity of the third Chern form for Griffiths positive bundles of arbitrary rank, and leaves the general case open.

Problem Statement

Finski's open problem (arXiv:2009.13107; J. London Math. Soc. 106 (2022), Introduction), at full generality with no dimension restriction. Let $V,W$ be complex vector spaces with $\dim V=\dim W=r$, and let $H:\mathrm{End}(V)\to\mathrm{End}(W)$ be a positive linear map, i.e. $H$ sends every nonzero positive semidefinite endomorphism of $V$ to a positive definite endomorphism of $W$. Let $D_V:\mathrm{End}(V)^{\otimes r}\to\mathbb{C}$ be the mixed discriminant, the polarization of the determinant, given in coordinates, for matrices $A_i=(a^i_{kl})_{k,l=1}^r$, by $D_V(A_1,\ldots,A_r)=\frac{1}{r!}\sum_{\sigma\in S_r}\det(a^{\sigma(i)}_{ik})$, and let $D_V^*:\mathbb{C}\to\mathrm{End}(V)^{\otimes r}$ be its dual map under the natural identification $\mathrm{End}(V)\simeq\mathrm{End}(V)^*$. Is the real number

$\Phi(H)=D_W\circ H^{\otimes r}\circ D_V^*$

always positive, i.e. $\Phi(H)>0$?

By Finski's Theorem 1.3 this is equivalent to Griffiths' 1969 question: for every nonzero nonnegative linear combination $P$ of Schur polynomials of weighted degree $k$, and every Griffiths positive Hermitian holomorphic vector bundle $(E,h)$ of rank $r$ over a complex manifold $X$ of dimension $n\ge k$, is the $(k,k)$-form $P(c_1(E,h),\ldots,c_r(E,h))$ weakly positive? The nonnegative variants ($H$ non-negative with $\Phi(H)\ge 0$; Griffiths semipositive bundles with weakly nonnegative forms) are the boundary versions of the same problem.

The cases $r\le 2$ (Finski 2022) and $r=3$ (Wan 2026) are settled affirmatively, so the open core is the general statement for $r\ge 4$, together with the equivalent residual instances of Griffiths' question (in particular all ranks $\ge 4$, and Schur forms of degree $\ge 4$ for rank-3 Griffiths positive bundles over manifolds of dimension $\ge 4$).

References

1. Siarhei Finski, On characteristic forms of positive vector bundles, mixed discriminants, and pushforward identities, Journal of the London Mathematical Society 106(2) (2022) 1539-1579, DOI 10.1112/jlms.12605, arXiv:2009.13107, https://arxiv.org/abs/2009.13107
2. Xueyuan Wan, Positivity of the third Chern form for Griffiths positive vector bundles, arXiv:2601.10424 (2026), https://arxiv.org/abs/2601.10424
3. Phillip A. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles, in Global Analysis (Papers in Honor of K. Kodaira), University of Tokyo Press, Tokyo, and Princeton University Press, Princeton, 1969, pp. 185-252; De Gruyter digital edition DOI 10.1515/9781400871230-011, https://doi.org/10.1515/9781400871230-011
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