Curvature of higher direct images

Abstract

Given a holomorphic family f : X → S of compact complex manifolds and a relatively ample line bundle L → X , the higher direct images R n−p f * Ω p X /S (L) carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images. This generalizes a result of Schumacher [Sch12], where he computed the curvature of R n−p f * Ω p X /S (K ⊗m X /S ) for a family of canonically polarized manifolds. For p = n, it coincides with a formula of Berndtsson obtained in [Be11]. Thus, when L is globally ample, we reprove his result [Be09] on the Nakano positivity of f * (K X /F ⊗ L).

Twisted Hodge filtration: Curvature of the determinant

“…First we recall the setting from [Na16]. Let f : X → S be a proper holomorphic submersion and (L, h) a line bundle on X .…”

“…We use the notation ψ l := ψ l for sections ψ l and write g dV = ω Xs /n!. The result from [Na16] is Theorem 1. Let f : X → S be a proper holomorphic submersion and (L, h) → X a relative ample line bundle.…”

“…The understanding of this situation has applications to moduli problems. An explicit curvature formula for these higher direct images is given in [Na16]. In general, the curvature of the intermediate higher direct images contains positive and negative contributions.…”

“…

Given a holomorphic family f : X → S of compact complex manifolds and a relative ample line bundle L → X , the higher direct images R n−p f * Ω p X /S (L) carry a natural hermitian metric. An explicit formula for the curvature tensor of these direct images is given in [Na16]. We prove that the determinant of the twisted Hodge filtration

…”

Notes on variation of Lefschetz star operator and $T$-Hodge theory

“…Remark: One may also prove the above theorem by a direct computation without using the Hodge star operator, see [25]. For other related results on the Lie-derivative connection, see [6], [14], [20], [21], [23], [24], [28], [29], [31].…”