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Locally Constant Fibrations and Positivity of Curvature

Niklas Müller

TL;DR

The paper establishes a sharp, structural bridge between curvature positivity and locally constant fibrations: up to finite étale covers, smooth projective X with nef $-K_X$ is a fibre bundle over a $K$-trivial base with locally constant transitions. It proves a converse: a projective locally constant fibration over a $K$-trivial base with nef $-K$ on the fibre inherits nefness for $-K_X$, and it extends the framework to singular spaces via klt theory. A central technical theme is the interplay between semistability, holomorphic connections, and numerical flatness of vector bundles, used to deduce flatness and local constancy via splitting of tangent sequences. The results yield a flexible construction principle for varieties with nef anti-canonical or nef tangent bundles and provide explicit criteria for projectivity and positivity, with applications to Albanese fibrations and the structure theory of nef tangent varieties.

Abstract

Up to finite étale cover, any smooth complex projective variety $X$ with nef anti-canonical bundle is a holomorphic fibre bundle over a $K$-trivial variety with locally constant transition functions. We show that this result is optimal by proving that any projective fibre bundle with locally constant transition functions over a $K$-trivial variety has a nef anti-canonical bundle. Moreover, we complement some results on the structure theory of varieties whose tangent bundle admits a singular hermitean metric of positive curvature.

Locally Constant Fibrations and Positivity of Curvature

TL;DR

The paper establishes a sharp, structural bridge between curvature positivity and locally constant fibrations: up to finite étale covers, smooth projective X with nef is a fibre bundle over a -trivial base with locally constant transitions. It proves a converse: a projective locally constant fibration over a -trivial base with nef on the fibre inherits nefness for , and it extends the framework to singular spaces via klt theory. A central technical theme is the interplay between semistability, holomorphic connections, and numerical flatness of vector bundles, used to deduce flatness and local constancy via splitting of tangent sequences. The results yield a flexible construction principle for varieties with nef anti-canonical or nef tangent bundles and provide explicit criteria for projectivity and positivity, with applications to Albanese fibrations and the structure theory of nef tangent varieties.

Abstract

Up to finite étale cover, any smooth complex projective variety with nef anti-canonical bundle is a holomorphic fibre bundle over a -trivial variety with locally constant transition functions. We show that this result is optimal by proving that any projective fibre bundle with locally constant transition functions over a -trivial variety has a nef anti-canonical bundle. Moreover, we complement some results on the structure theory of varieties whose tangent bundle admits a singular hermitean metric of positive curvature.
Paper Structure (12 sections, 19 theorems, 51 equations)

This paper contains 12 sections, 19 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Selected facts on Semistability and Flatness
  4. Some words on Principal Bundles
  5. Projectivity of Locally Constant Fibre Bundles
  6. Fibrations and Splittings of the relative Tangent Sequence
  7. Step 1: Existence of a Relatively Ample and Flat Line Bundle
  8. Step 2: Existence of a Holomorphic Connection
  9. Step 3: Conclusion of the Proof
  10. Locally Constant Fibrations and Positivity of Curvature
  11. Varieties with Nef Anti-Canonical Bundle
  12. Varieties with Nef Tangent Bundle

Key Result

Theorem 1.1

Let $Y$ be a smooth projective variety with $\mathop{\mathrm{c}}\nolimits_1(Y) = 0$, let $F$ be a smooth, projective, rationally connected variety whose anti-canonical divisor $-K_F$ is nef and let $\rho\colon \pi_1(Y) \rightarrow \mathop{\mathrm{Aut}}\nolimits(F)$ be any group homomorphism. Write where we denote by $\widetilde{Y}\rightarrow Y$ the universal cover of $Y$ and where the action of $

Theorems & Definitions (47)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 37 more