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A relative Nadel-type vanishing theorem

Jingcao Wu

TL;DR

This work extends Nadel-type vanishing to a relative setting for fibrations $f:X\to Y$ by combining the relative asymptotic multiplier ideal $\mathscr{I}(f,\|L\|)$ with Nakano semi-positivity (and its strong singular analogue) to show vanishing of pushforwards $R^{q}f_{\ast}(K_{X}\otimes L\otimes E\otimes\mathscr{I}(f,\|L\|))$ for $q>l-\kappa(L,f)$. It also proves absolute and singular versions via the Monge–Ampère method and strong openness, yielding $H^{q}(X,K_{X}\otimes L\otimes E\otimes\mathscr{I}(\|L\|))=0$ or $H^{q}(X,K_{X}\otimes L\otimes E\otimes\mathcal{E}(h))=0$ for $q>n-\cdot$ where appropriate. A key technical backbone is a Mat22-style injectivity result that ensures torsion-freeness of the higher direct images, enabling extension from fibers and base change arguments. The paper then carries these ideas through a dimension-inductive strategy to treat $f$-nef and analytic almost base point free $L$, achieving the stated relative vanishing theorems and their corollaries for pushforwards along a fibration in the Kähler and projective settings.

Abstract

Let $f:X\rightarrow Y$ be a Kähler fibration from a complex manifold $X$ to an analytic space $Y$. We show several relative Nadel-type vanishing theorems.

A relative Nadel-type vanishing theorem

TL;DR

This work extends Nadel-type vanishing to a relative setting for fibrations by combining the relative asymptotic multiplier ideal with Nakano semi-positivity (and its strong singular analogue) to show vanishing of pushforwards for . It also proves absolute and singular versions via the Monge–Ampère method and strong openness, yielding or for where appropriate. A key technical backbone is a Mat22-style injectivity result that ensures torsion-freeness of the higher direct images, enabling extension from fibers and base change arguments. The paper then carries these ideas through a dimension-inductive strategy to treat -nef and analytic almost base point free , achieving the stated relative vanishing theorems and their corollaries for pushforwards along a fibration in the Kähler and projective settings.

Abstract

Let be a Kähler fibration from a complex manifold to an analytic space . We show several relative Nadel-type vanishing theorems.
Paper Structure (8 sections, 18 theorems, 103 equations)

This paper contains 8 sections, 18 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Asymptotic multiplier ideal sheaf
  4. Singular Hermitian metrics
  5. A Nadel-type vanishing theorem
  6. Main theorem
  7. Technical preparation
  8. Proof of the main theorem

Key Result

Theorem 1.1

Let $f:X\rightarrow Y$ be a surjective projective morphism of quasi-projective varieties, with $X$ non-singular. Denote by $l$ the dimension of a general fiber of $f$. Let $L$ be a holomorphic line bundle on $X$ with $\kappa(L,f)=l$. Then for $q>0$.

Theorems & Definitions (34)

  • Theorem 1.1: (c.f. Laz04, Generalizations 9.1.22 and 11.2.15)
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • ...and 24 more