Universal vector bundles, push-forward formulae and positivity of characteristic forms

Technical Summary of "Universal Vector Bundles, Push-Forward Formulæ and Positivity of Characteristic Forms"

Problem Statement
The paper addresses the relationship between the cohomological and differential-geometric levels of characteristic classes for holomorphic vector bundles. Specifically, given a Hermitian holomorphic vector bundle $(E, h)$ over a complex manifold $X$, one can consider flag bundles $\pi_\rho: F_\rho(E) \to X$ associated with a sequence of dimensions. These bundles carry universal vector bundles (tautological sub-bundles and their quotients) endowed with induced Hermitian metrics.

A classical problem in algebraic geometry involves determining a universal formula (Gysin or push-forward formula) to express the push-forward of a polynomial in the Chern classes of these universal bundles as a polynomial in the Chern classes of the original bundle $E$. While such formulas are well-established in cohomology, the paper investigates the analogue at the level of differential forms (Chern–Weil forms). The central question is whether the push-forward of a polynomial in the Chern forms of the universal bundles (computed via integration along the fibers) coincides pointwise with the polynomial in the Chern forms of $E$ obtained by applying the known cohomological Gysin formula. Prior to this work, it was generally expected that these two forms might differ by an exact form.

Methodology
The author employs a pointwise differential-geometric approach, relying on explicit local computations of curvature tensors. The methodology proceeds as follows:

1. Local Frame Construction: The paper constructs specific local holomorphic coordinates and frames for the universal vector bundles over the flag bundle $F_\rho(E)$. These frames are adapted to the tautological filtration and the induced Hermitian metrics.
2. Explicit Curvature Computation: A critical technical step is the explicit computation of the Chern curvature tensor of the universal vector bundles (both tautological sub-bundles and their quotients) at a point. The author proves that the curvature of a universal quotient bundle $U_{\rho,l}/U_{\rho,\ell}$ can be decomposed into a "horizontal" part (related to the pullback of the curvature of $E$) and a "vertical" part (intrinsic to the fiber geometry).
3. Decomposition of Curvature: The curvature is expressed as the sum of a horizontal tensor $\theta^{(\ell,l)}$ (which depends on the curvature of $E$) and a vertical tensor $\omega^{(\ell,l)}$. This decomposition allows the Chern forms of the universal bundles to be written as polynomials in the entries of the curvature of $E$ and the vertical differentials.
4. Integration and Universality: By analyzing the structure of these polynomials, the author demonstrates that the push-forward operation (integration along the fibers) acts on the vertical components to produce coefficients that are universal constants (integrals over the flag manifold). The horizontal components, which depend on the base manifold $X$, are preserved.
5. Comparison with Cohomology: The resulting expression is compared against the known cohomological Gysin formula. The author utilizes a universality argument, testing the difference between the differential form result and the cohomological prediction on a specific class of split vector bundles (direct sums of line bundles with positive curvature) to show that the difference must vanish identically.

Key Contributions and Results

• Theorem A (Pointwise Push-Forward Identity): The main result (Theorem 3.6) establishes that for any homogeneous polynomial $F$ in the Chern forms of all universal vector bundles over a flag bundle, the push-forward via integration along the fibers equals the polynomial $\Phi$ (determined by the cohomological Gysin formula) evaluated directly on the Chern forms of the original bundle $(E, h)$.
  $$(\pi_\rho)_* F(c^\bullet(E_1, H_1), \dots, c^\bullet(E_N, H_N)) = \Phi(c^\bullet(E, h))$$
  This equality holds pointwise as differential forms, implying that the "exact form" difference is zero.

• Explicit Curvature Formulas: Theorem 2.1 provides explicit formulas for the Chern curvature of universal vector bundles on flag bundles in terms of the curvature of the base bundle and vertical differentials. This generalizes previous results known for projective bundles and Grassmann bundles.

• Refinement of the Jacobi–Trudi Identity: As an application, the paper provides an alternative version of the Jacobi–Trudi identity at the level of differential forms (Proposition 4.2). It shows that every Schur form (a Schur polynomial evaluated in Chern forms) can be obtained as a push-forward from the complete flag bundle, refining previous results on the cohomological level.

• Positivity of Characteristic Forms: The paper investigates the positivity of characteristic forms for Griffiths semipositive vector bundles. It confirms Griffiths' conjecture (which posits that non-negative combinations of Schur polynomials evaluated in Chern forms yield positive differential forms) for a specific family of forms.

  • Theorem 4.5: It proves that the push-forward of forms like $c_1(Q_s, h_s)^\alpha \wedge c_2(Q_s, h_s)^\beta$ from Grassmann bundles $G_s(E)$ to $X$ yields positive differential forms on $X$ that belong to the positive convex cone spanned by Schur forms.
  • This result provides new evidence for Griffiths' conjecture, covering cases not previously known, such as specific combinations involving the second Chern form in higher ranks.

Significance and Claims
The paper claims that its primary significance lies in bridging the gap between cohomological formulas and their differential-geometric representatives. By proving that the push-forward of Chern forms commutes with the application of universal Gysin formulas, the author eliminates the need for explicit fiber integration to determine characteristic forms on the base manifold. One can simply apply the known cohomological formulas to the Chern forms.

Regarding positivity, the paper modestly claims to partially confirm Griffiths' conjecture. It establishes the positivity of a new family of characteristic forms derived from Grassmann bundles. The author notes that while these forms are positive, they are not necessarily "strongly positive" (in the sense of being sums of squares of $(2,2)$-forms) due to the nature of the push-forward involving squares of $(2,2)$-forms. The work distinguishes the cone of forms generated by these push-forwards from cones generated by other methods (such as those in [DF22]), showing that neither is contained within the other, thereby expanding the known landscape of positive characteristic forms for Griffiths semipositive bundles.

The results are presented as a direct consequence of the explicit curvature computations and the universality of the push-forward operation, without introducing new experimental proposals or unverified future implications beyond the scope of the established theorems.