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Some remarks on fibrations in complex geometry

Nobuhiro Honda, Jeff Viaclovsky

TL;DR

This paper analyzes holomorphic fibrations in complex geometry, focusing on fiber structure, discriminant loci, and monodromy, and proves a naturality property of the Leray spectral sequence under open inclusions. It derives a cohomological bound b^1(Z_y) ≤ b^1(F) for fibers over curves, with equality characterized by trivial local monodromy, and extends to conic bundles over surfaces with rational tree singularities, obtaining b^1(Z_y)=0 for all y and Euler-characteristic bounds under equidimensionality. It introduces rational tree surface singularities and leverages Leray naturality to relate discriminant geometry with cohomology and local invariant cycles, including a non-Kähler example illustrating the limits of LIC in higher dimensions. The work provides a cohesive framework linking fibration geometry, singularities, and cohomological invariants, with explicit theorems and a thorough naturality analysis of Leray sequences.

Abstract

In this article, we discuss some properties of holomorphic fibrations in the complex analytic setting.

Some remarks on fibrations in complex geometry

TL;DR

This paper analyzes holomorphic fibrations in complex geometry, focusing on fiber structure, discriminant loci, and monodromy, and proves a naturality property of the Leray spectral sequence under open inclusions. It derives a cohomological bound b^1(Z_y) ≤ b^1(F) for fibers over curves, with equality characterized by trivial local monodromy, and extends to conic bundles over surfaces with rational tree singularities, obtaining b^1(Z_y)=0 for all y and Euler-characteristic bounds under equidimensionality. It introduces rational tree surface singularities and leverages Leray naturality to relate discriminant geometry with cohomology and local invariant cycles, including a non-Kähler example illustrating the limits of LIC in higher dimensions. The work provides a cohesive framework linking fibration geometry, singularities, and cohomological invariants, with explicit theorems and a thorough naturality analysis of Leray sequences.

Abstract

In this article, we discuss some properties of holomorphic fibrations in the complex analytic setting.

Paper Structure

This paper contains 16 sections, 23 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Holomorphic fibrations
  3. Naturality of Leray spectral sequence
  4. Acknowledgements
  5. Holomorphic fibrations
  6. Fiber dimension
  7. Discriminant locus
  8. Monodromy
  9. Ordinary and non-ordinary points
  10. Fibrations over a curve
  11. Deformation retractions
  12. Proof of Theorem \ref{['t:t1']}
  13. Local invariant cycles theorem
  14. Surface singularities
  15. Conic bundle over a surface
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $f : Z \rightarrow C$ be a fibration, where $Z$ is a compact complex $n$-manifold and $C$ is a smooth curve. Then for all$y \in C$, where $F$ is the general fiber. Equality holds in b1est for some $y \in C$ if and only if the local monodromy $\rho^{(1)}_y$ around $y$ is trivial. If $n = 2$, then we have the inequality with equality if and only if the global monodromy $\rho^{(1)}$ is trivial

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 41 more