Double Mixed Discriminant in Complex Geometry

• Double Mixed Discriminant is a multilinear algebraic invariant derived via polarization of determinants, encoding interactions among endomorphisms in complex vector spaces.
• It plays a critical role in establishing the positivity of Schur forms in rank-three Hermitian holomorphic bundles by linking determinant-like functionals to curvature properties.
• Analytical tools such as integral representations over the complex unit sphere and algebraic inequalities support its application in resolving conjectures like Griffiths' positivity and Finski’s problem.

The double mixed discriminant is a multilinear algebraic invariant central to the analysis of Schur forms in the context of positive Hermitian holomorphic vector bundles, especially in connection with Finski's problem and the Schur positivity conjecture of Griffiths in rank three. Its canonical role is as the determinant-like functional encoding the interaction between multiple endomorphisms, with profound implications for positivity phenomena in complex differential geometry.

1. Formal Definition and Construction

Let $V$ and $W$ be complex vector spaces of dimension $3$. Given a linear map $H: \operatorname{End}(V) \to \operatorname{End}(W)$, the double mixed discriminant arises through the multilinear polarization of the determinant, known as the mixed discriminant $D: \operatorname{End}(W)^{\times 3} \to \mathbb{C}$. In coordinates, for $$A^i = (a^i_{p\bar q})_{1 \le p,q \le 3} \in \operatorname{End}(W)$$, the mixed discriminant is

$$D(A^1, A^2, A^3) = \frac{1}{3!}\sum_{\sigma \in S_3} \det(a^{\sigma(i)}_{\,i\bar k})_{i,k=1}^3$$

which can equivalently be expressed via traces as

$$6 D(U,V,W) = \operatorname{tr}(U) \operatorname{tr}(V) \operatorname{tr}(W) - \operatorname{tr}(U) \operatorname{tr}(VW) - \operatorname{tr}(V) \operatorname{tr}(UW) - \operatorname{tr}(W)\operatorname{tr}(UV) + \operatorname{tr}(UVW) + \operatorname{tr}(UWV).$$

For the "double" construction, let $B_{i\bar j} = H(E_{i\bar j}) \in M_3(\mathbb{C})$, where $E_{i\bar j}$ is the standard matrix unit. The dual mixed discriminant $$\mathcal{D}_V^* : \mathbb{C} \to \operatorname{End}(V)^{\otimes 3}$$ is defined as

$$\mathcal{D}_V^*(1) = \frac{1}{3!}\sum_{\sigma, \tau \in S_3} \operatorname{sgn}(\sigma)\,\operatorname{sgn}(\tau)\; E_{\sigma(1)\, \overline{\tau(1)}} \otimes E_{\sigma(2)\, \overline{\tau(2)}} \otimes E_{\sigma(3)\, \overline{\tau(3)}}.$$

Then the double mixed discriminant is

$$\Phi = \mathcal{D}_W \circ H^{\otimes 3} \circ \mathcal{D}_V^*(1) = \sum_{\sigma \in S_3} \operatorname{sgn}(\sigma) D(B_{1\overline{\sigma(1)}}, B_{2\overline{\sigma(2)}}, B_{3\overline{\sigma(3)}}).$$

This formula encodes the interaction of the linear map $H$ with the algebraic structure of $V$ and $W$ via their endomorphism spaces (Wan, 15 Jan 2026).

2. Relation to Schur Forms and Chern Theory

Schur forms $s_\lambda(E,h)$, indexed by partitions $\lambda$ of degree $k$ and constructed as polynomials $P_\lambda$ in Chern forms, can be written as

$$s_\lambda(E,h) = P_\lambda(c(E,h)) = \det( c_{\lambda_i - i + j}(E,h) )_{1 \leq i,j \leq r}$$

where $$c(E,h) = \det(I_E + \frac{\sqrt{-1}}{2\pi} R^E) = \sum_{i=0}^r c_i(E,h)$$ for a Hermitian holomorphic vector bundle $(E,h)$ of rank $r$ (Wan, 15 Jan 2026, Wan, 2023).

Specifically, when $r=3$ and $k=3$ (total degree), there are exactly three nontrivial Schur forms:

• $s_{(3,0,0)} = c_3(E,h)$,
• $s_{(2,1,0)} = c_1(E,h) \wedge c_2(E,h) - c_3(E,h)$,
• $$s_{(1,1,1)} = c_1(E,h)^3 - 2c_1(E,h)\wedge c_2(E,h) + c_3(E,h)$$.

By Finski’s pushforward formula, the top Chern form $c_3(E,h)$ is identified with the double mixed discriminant $\Phi$ formed from the curvature endomorphisms of $E$. Consequently, positivity of $\Phi$ gives the required weak positivity of $c_3(E,h)$, and hence controls all rank-three Schur positivity phenomena (Wan, 15 Jan 2026).

3. Positivity Theorems and Griffiths–Finski Correspondence

Finski established that, for fixed rank $r$, Griffiths' conjecture on the weak positivity of Schur forms is equivalent to the positivity of the operator $$\mathcal{D}_W \circ H^{\otimes r} \circ \mathcal{D}_V^*$$. For $r=3$, the statement reduces to the positivity of the double mixed discriminant $\Phi$ under the following assumptions on the blocks $B_{i\bar j}$:

• Griffiths-type positivity: $$\sum_{i,j} B_{i\bar j} \xi^i \overline{\xi^j} \succ 0$$ for all $\xi \neq 0$.
• Doubly-stochastic normalization: $\sum_{i=1}^3 B_{i\bar i} = 3I$ and $\operatorname{tr}(B_{i\bar j}) = 3\delta_{ij}$ for $1 \leq i,j \leq 3$.

Under these conditions, one has $\Phi > 0$ (Wan, 15 Jan 2026).

4. Integral Representations and Algebraic Inequalities

An analytic approach to establishing $\Phi > 0$ uses integral representations over the complex unit sphere $S^5 \subset \mathbb{C}^3$. Define $C(\xi) = (\xi^* B_{i\bar j} \xi)_{i,j=1}^3$ for $\xi \in S^5$. With $\lambda_1, \lambda_2, \lambda_3$ the eigenvalues of $C(\xi)$, and $$\sigma_2(C) = \lambda_1\lambda_2 + \lambda_1\lambda_3 + \lambda_2\lambda_3$$, it is shown that

$$\Phi = \int_{S^5} \left( 10\, \det C(\xi) + 27 - 12\,\sigma_2(C(\xi)) \right)\, d\mu(\xi)$$

where $d\mu$ is the invariant measure. A classical Schur-inequality argument,

$$(\lambda_1 + \lambda_2 + \lambda_3)^3 + 9\lambda_1\lambda_2\lambda_3 \ge 4(\lambda_1 + \lambda_2 + \lambda_3)\sigma_2(C),$$

with $\lambda_1 + \lambda_2 + \lambda_3 = 3$, shows the integrand is nonnegative and, under positivity assumptions, strictly positive; thus $\Phi > 0$ (Wan, 15 Jan 2026).

5. Examples, Special Cases, and Limitations

In rank $r=2$, a similar construction shows

$$\Phi = D(B_{1\bar 1}, B_{2\bar 2}) - D(B_{1\bar 2}, B_{2\bar 1}) \ge 1$$

whenever the analogous positivity conditions are satisfied (Wan, 15 Jan 2026). The positivity of the double mixed discriminant in rank two was previously known, but the $3 \times 3$ case, resolved in (Wan, 15 Jan 2026), settles Finski’s open problem for rank/dimension three, completing the answer to Griffiths' 1969 question in this case.

A notable limitation is that the "integral-sphere" method underlying the analysis of $\Phi$ fails in rank four due to the appearance of a residual term not symmetric in the eigenvalues, and so extension to $r \ge 4$ remains an open problem (Wan, 15 Jan 2026).

6. Broader Context and Applications

The double mixed discriminant plays a critical role in the interface of algebraic geometry, complex differential geometry, and multilinear algebra:

• It provides the algebraic mechanism underlying the positivity (weak, or strong under further curvature hypotheses) of Schur forms or generalized Chern forms in the study of Hermitian holomorphic vector bundles (Wan, 2023, Wan, 15 Jan 2026).
• The correspondence between positivity phenomena (Nakano, dual-Nakano, decomposable positivity, Griffiths positivity) and the sign of the double mixed discriminant links geometric problems to explicit multilinear algebraic inequalities (Wan, 15 Jan 2026).
• Analytic techniques such as operator scaling (Gurvits–Sinkhorn) allow normalization of the relevant endomorphism arrays without affecting the sign of $\Phi$, facilitating the proof strategy (Wan, 15 Jan 2026).
• In the representation-theoretic setting, the mixed discriminant and its double appear in the context of dualities, multilinear invariants, and pushforward formulas for characteristic classes.

A plausible implication is that, as new ideas emerge for $r \geq 4$, extensions of the double mixed discriminant will continue to act as obstructions or validating functionals for generalized positivity conjectures in geometric analysis and algebraic geometry.