𝐿²-metrics, projective flatness and families of polarized abelian varieties

Abstract

Abstract. We compute the curvature of the L 2 -metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the L 2 -metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space. §0. Introduction It is well known that for a family π : A → S of principally polarized abelian varieties, the theta bundles on the fibers of π form a line bundle Θ over A. From the fact that the theta characters of level n, n ∈ Z + , satisfy a certain heat equation, one knows that the direct image vector bundles π * Θ ⊗n admit projectively flat connections such that the theta characters are parallel sections. Moreover, from the fact that the theta characters form an orthonormal basis with respect to a certain natural L 2 -pairing, one knows that the above projectively flat connections are indeed Hermitian connections (cf. [APW, §5]).As a generalization of the above, one considers in conformal field theory a family of Riemann surfaces and an associated family π : M → S of moduli spaces of stable vector bundles over the Riemann surfaces. As before, the so-called generalized theta bundles over the fibers of π form a line bundle Θ over M. An important result in conformal field theory in [TUY], [H] and [APW] was to show that for n ∈ Z + , π * Θ ⊗n admits a projectively flat connection, whose construction depends on the fact that certain generalized heat equations, or KZ equations, hold for sections of generalized theta bundles.In another direction, instead of just considering the theta bundles, Welters [W], following [M1], showed that for a family of polarized abelian varieties π : A → S and any relatively ample line bundle L over A, π * L always admits a projectively flat connection. This was established from hypercohomological considerations in [W], which led to certain generalized heat equations for sections of L (cf. also [H]).However the projectively flat connections constructed in [H] and [W] are not Hermitian connections in general. From the viewpoint of representation theory, a vector bundle of rank r over S admits a projectively flat connection (resp. projectively flat Hermitian connection) if and only if E arises from a representation of

An alternative approach on the singular metric on a vector bundle

“…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.…”

Curvature of higher direct images

“…-If ϕ = s and ψ = t are just sections of a vector bundle, the Lie derivative is nothing else but the covariant derivative. This is used in [14] in their computation of the curvature. The preceding proposition…”

Curvature of higher direct images

“…If ϕ = s and ψ = t are just sections of a vector bundle, the Lie derivative is nothing else but the covariant derivative. This is used in [TW04] in their computation of the curvature. The preceding proposition means that the Lie derivative of the metric h vanishes, which just means that the covariant derivative vanishes.…”