[Paper Review] A transcendental approach to Kollár's injectivity theorem II

Osamu Fujino|arXiv (Cornell University)|May 9, 2007

Geometry and complex manifolds21 references17 citations

TL;DR

This paper establishes a relative version of Kollár's injectivity theorem using a transcendental approach based on curvature conditions and the Ohsawa–Takegoshi twisted Nakano identity. It proves that under Nakano semi-positivity and curvature bounds on line bundles, multiplication by a holomorphic section induces an injective map on higher direct images of canonical bundles twisted by multiplier ideal sheaves.

ABSTRACT

We treat a relative version of the main theorem in my previous paper: A transcendental approach to Kollár's injectivity theorem. More explicitly, we give a curvature condition that implies Kollár type cohomology injectivity theorems in the relative setting. To carry out this generalization, we use the Ohsawa-Takegoshi twisted version of Nakano's identity.

Motivation & Objective

• To generalize Kollár's injectivity theorem to the relative setting using analytic methods.
• To establish cohomological injectivity theorems under curvature conditions on holomorphic bundles with singular metrics.
• To provide a framework for torsion-freeness and vanishing theorems in complex geometry via curvature-based criteria.
• To extend Enoki's injectivity theorem and Kawamata–Viehweg–Nadel vanishing to relative morphisms using multiplier ideal sheaves.
• To clarify the relationship between geometric and analytic approaches to vanishing theorems in algebraic geometry.

Proposed method

• Utilizes the Ohsawa–Takegoshi twisted version of Nakano's identity to analyze $\bar{\partial}$-equations on vector bundles.
• Applies curvature conditions: Nakano semi-positivity of $\Theta(E) + \mathrm{Id}_E \otimes \Theta(F)$ and $\Theta(F) \geq -\widetilde{\gamma}$ in the sense of currents.
• Imposes a strict positivity condition involving $\varepsilon_0 \mathrm{Id}_E \otimes \Theta(L)$ to ensure injectivity.
• Employs multiplier ideal sheaves $\mathcal{J}(h_F)$ associated to singular hermitian metrics on line bundles.
• Analyzes the induced map $\times s: R^q f_*(K_X \otimes E \otimes F \otimes \mathcal{J}(h_F)) \to R^q f_*(K_X \otimes E \otimes F \otimes \mathcal{J}(h_F) \otimes L)$ for holomorphic sections $s$.
• Uses the relative setting via proper surjective morphisms $f: X \to Y$ between complex varieties and Kähler manifolds.

Experimental results

Research questions

• RQ1 Under what curvature conditions does the multiplication by a holomorphic section induce injectivity on higher direct images in the relative setting?
• RQ2 How can Nakano semi-positivity and singular metric curvature bounds be combined to ensure torsion-freeness of direct image sheaves?
• RQ3 What is the precise relationship between Kollár's geometric injectivity theorem and Enoki's analytic version in the relative case?
• RQ4 Can the Ohsawa–Takegoshi twisted Nakano identity be used to derive new vanishing theorems for multiplier ideal sheaves?
• RQ5 Are there examples where a line bundle is nef and big but not semi-ample, and how does this affect cohomological injectivity?

Key findings

• The main theorem establishes injectivity of the multiplication map $\times s$ on $R^q f_*(K_X \otimes E \otimes F \otimes \mathcal{J}(h_F))$ under curvature and metric conditions.
• Torsion-freeness of $R^q f_*(K_X \otimes E \otimes F \otimes \mathcal{J}(h_F))$ is proven for all $q \geq 0$, with the consequence that the sheaf vanishes for $q > \dim X - \dim Y$.
• A Kawamata–Viehweg–Nadel type vanishing theorem is obtained: $R^q f_*(K_X \otimes E \otimes \mathcal{L} \otimes \mathcal{J}) = 0$ for $q > 0$, under $f$-nef-big and Nakano semi-positive conditions.
• A Kollár-type vanishing theorem is derived: $R^p g_* R^q f_*(K_X \otimes E \otimes \mathcal{L} \otimes \mathcal{J}) = 0$ for $p > 0$, $q \geq 0$, when $\mathcal{L}^\otimes m \simeq f^*\mathcal{N} \otimes \mathcal{O}_X(D)$ with $\mathcal{N}$ $g$-nef-big.
• Examples are constructed showing that semi-positivity does not imply semi-ampleness, and that injectivity can fail even when $\mathcal{M} \cdot C' > 0$ for all curves $C'$.
• The paper leaves open whether certain line bundles with positive intersection but no sections (e.g., in Example 5.9) admit smooth hermitian metrics with semi-positive curvature.