On characteristic forms of positive vector bundles, mixed discriminants, and pushforward identities

Abstract

We prove that Schur polynomials in Chern forms of Nakano and dual Nakano positive vector bundles are positive as differential forms. Moreover, modulo a statement about the positivity of a "double mixed discriminant" of linear operators on matrices, which preserve the cone of positive definite matrices, we establish that Schur polynomials in Chern forms of Griffiths positive vector bundles are weakly-positive as differential forms. This provides differential-geometric versions of Fulton-Lazarsfeld inequalities for ample vector bundles.An interpretation of positivity conditions for vector bundles through operator theory is in the core of our approach. Another important step in our proof is to establish a certain pushforward identity for characteristic forms, refining the determinantal formula of Kempf-Laksov for homolorphic vector bundles on the level of differential forms. In the same vein, we establish a local version of Jacobi-Trudi identity.Then we study the inverse problem and show that already for vector bundles over complex surfaces, one cannot characterize Griffiths positivity (and even ampleness) through the positivity of Schur polynomials, even if one takes into consideration all quotients of a vector bundle.

Positivity of Schur forms for strongly decomposably positive vector bundles

“…In this section, we recall positivity notions for differential forms. For more details, one can refer to [7, Section 1.1] and [15,9]. Let V be a complex vector space of dimension n and let 𝑛 ) the dual basis of 𝑉 * .…”

“…From the above definition, a (dual) Nakano positive vector bundle must be decomposably positive. From [9, Proposition 2.21], for decomposable positivity is equivalent to Griffiths positivity; that is, for any non-zero and . Decomposable positivity is strictly stronger than Griffiths positivity for all other .…”

“…Li [17, Proposition 3.1] extended Bott-Chern’s result and obtained all Schur forms are non-negative. Later, Finski [9, Theorem 2.15] proved the equivalence of Bott-Chern non-negativity and dual Nakano non-negativity. Moreover, using a purely algebraic method, Finski [9, Section 3.4] proved that all Schur forms of a Nakano non-negative vector bundle are non-negative.…”

“…Later, Finski [9, Theorem 2.15] proved the equivalence of Bott-Chern non-negativity and dual Nakano non-negativity. Moreover, using a purely algebraic method, Finski [9, Section 3.4] proved that all Schur forms of a Nakano non-negative vector bundle are non-negative. For (dual) Nakano positive vector bundles, Finski [9, Theorem 1.1] proved that all Schur forms are positive by the refinement of the determinantal formula of Kempf-Laksov on the level of differential forms.…”

“…Moreover, using a purely algebraic method, Finski [9, Section 3.4] proved that all Schur forms of a Nakano non-negative vector bundle are non-negative. For (dual) Nakano positive vector bundles, Finski [9, Theorem 1.1] proved that all Schur forms are positive by the refinement of the determinantal formula of Kempf-Laksov on the level of differential forms. However, as pointed out by Finski [9, Remark 3.18], the above algebraic method can be used to deal with the case of non-negativity, while for the positivity statement, it is not clear if one can refine the algebraic method because there is no similar criterion for (dual) Nakano positivity (see [9, Remark 2.16]) and there is little known about the specific structure of the forms defined in [9, (3.83)].…”