Approximations and examples of singular Hermitian metrics on vector bundles

Abstract

Abstract. We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic curves.

Nadel–Nakano vanishing theorems of vector bundles with singular Hermitian metrics

“…Although a singular Hermitian metric of a vector bundle has been investigated in many papers (for example [3,14,15,19,20]), there exist few results on vanishing theorems for vector bundles with singular Hermitian metrics. We explain the details of the investigations of a singular Hermitian metric of a vector bundle below.…”

“…We do not know if E(h), unlike a multiplier ideal sheaf, is coherent, In [5], de Cataldo proved that E(h) is coherent and a Nadel-Nakano type vanishing theorem if h has an approximate sequence of smooth Hermitian metrics {h µ } satisfying h µ ↑ h pointwise and √ −1Θ E,hµ − ηω ⊗ Id E 0 in the sense of Nakano for some positive and continuous function η. However, h does not always have such an approximate sequence (see [15,Example 4.4]). Therefore these problems are open.…”

A converse of Hörmander’s L2-estimate and new positivity notions for vector bundles

“…For general vector bundles, little is known about the coherence of such sheaves, because of the lack of L 2 -estimates for general singular Hermitian metrics. The first author has proved the coherence of such sheaves for singular Hermitian metrics induced by holomorphic sections [H,Theorem 1.1]. We prove that the sheaf of locally square integrable holomorphic sections with respect a metric which is positive in the sense of twisted Hörmander is coherent.…”

“…Since coherence is a local property, we can assume that Ω is a bounded domain in C n , E = Ω × C r is the trivial bundle over Ω, and each element of h ⋆ is bounded on Ω. Let H 0 (2,h) (Ω, C r ) be the square integrable C r -valued holomorphic functions with respect to h on Ω. By the strong Noetherian property of coherent sheaves, H 0 (2,h) (Ω, C r ) generates a coherent ideal sheaf F ⊂ O(E) = O(C r ).…”

Metric Properties of Parabolic Ample Bundles

“…It is but natural to wonder if the same kind of a conjecture can be made for singular Hermitian metrics. Unfortunately, the notion of a singular Hermitian metric on general vector bundles (as opposed to line bundles where a lot of work has been done) is quite subtle and only recently has there been progress on it [4,5,19,26,29,38,41,42]. A compromise can be made by choosing to work with parabolic bundles, which are essentially vector bundles equipped with flags (and weights) over divisors.…”