An approach to Griffiths conjecture

Abstract

The Griffiths conjecture asserts that every ample vector bundle E over a compact complex manifold S admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on O P(E * ) (1) to induce a Griffiths positive L 2 -metric on the vector bundle E. This result suggests to study the relative Kähler-Ricci flow on O P(E * ) (1) for the fibration P(E * ) → S. We define a flow and give arguments for the convergence.

An alternative approach on the singular metric on a vector bundle

“…The opposite direction is mainly based on the technique developed in [21]. Let's explain the result in [21] first.…”

“…The opposite direction is mainly based on the technique developed in [21]. Let's explain the result in [21] first. In fact, it is proved in [21] that given a metric ϕ on O E (1) with positive curvature, if it induces an isometry on (1), then the L 2 -metric h on E defined by ϕ is positive in the sense of Griffiths.…”

“…Let's explain the result in [21] first. In fact, it is proved in [21] that given a metric ϕ on O E (1) with positive curvature, if it induces an isometry on (1), then the L 2 -metric h on E defined by ϕ is positive in the sense of Griffiths. More precisely, [21] actually indicates that (8) iΘ h α βi j w α wβ dz i ∧ dz j = −Ψ T h up to a constant.…”

“…In fact, it is proved in [21] that given a metric ϕ on O E (1) with positive curvature, if it induces an isometry on (1), then the L 2 -metric h on E defined by ϕ is positive in the sense of Griffiths. More precisely, [21] actually indicates that (8) iΘ h α βi j w α wβ dz i ∧ dz j = −Ψ T h up to a constant. Hence the Griffiths positivity of h is due to the Kobayashi negativity of Ψ, which is equivalent to the ampleness of ϕ by formula (2).…”

“…In order to complete the proof under the same scheme as [21], we only need to prove that the formula ( 8) is suitable for our situation. It is a byproduct of [21] except that we prefer to use Berndtsson's curvature formula in [1] other than the To-Wang formula [28]. The former one seems to be more suitable for our situation.…”