Representability of Chern–Weil forms

Abstract

Abstract. In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.

Metric Properties of Parabolic Ample Bundles

“…(Our proof is a calculation involving generating functions and is hence technically slightly different from the analogous ones for usual bundles by [21] and [24].) In addition, for surfaces, akin to [39] we prove that parabolic ample stable bundles (arising from good Kawamata covers) admit metrics whose Schur forms are positive.…”

“…This would complete the proof of Theorem 5.8. Indeed, just as in [39], is admissible and satisfies the desired properties.…”

“…In [10] a result akin to Bloch-Gieseker [15] was proven for parabolic Chern classes of parabolic ample bundles. Therefore, If we manage to do so, then the calculations in [39] show that c 2 ( G) > 0. This would complete the proof of Theorem 5.8.…”

“…Our proof has a small technical innovation in terms of generating functions and we hope it generalizes to computing pushforwards for flag bundles. Lastly, we prove a parabolic version of a result (for parabolic structures arising from good Kawamata covers) in [39] concerning the existence of metrics whose Schur forms are positive on stable bundles over surfaces. (4) In Section 6 we prove a parabolic analog of Berndtsson's theorem (as above, for parabolic structures arising from good Kawamata covers), that is, if E is Hartshorne ample, E ⊗ det(E) is Nakano positive.…”

“…Guler [24] and Diverio [21] proved, using a complicated calculation based on an elegant idea of Guler, that the signed Segre forms of Griffiths positive bundles are positive (in particular, on surfaces the Schur polynomials of a Griffiths positive metric are positive pointwise). In [39], it was proven that a Hartshorne ample semistable bundle over a surface admits a metric whose Schur polynomials are positive pointwise. It is still unknown as to whether Schur polynomials of Griffiths positive metrics are pointwise positive; however if so, this would be further indirect evidence.…”