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Bi-Hermitian and locally conformally Kähler surfaces

Massimiliano Pontecorvo

TL;DR

We address the problem of identifying compact complex surfaces that admit bi-Hermitian structures—two integrable, $g$-orthogonal complex structures with the same orientation—while clarifying their relationship with locally conformally Kähler (LCK) geometry. The approach builds the fundamental divisor $T$ and the fundamental flat bundle $F$, analyzes Lee forms and Gauduchon degrees, and stratifies surfaces by $b_1$ and Kodaira class; key results connect LCK data to bi-Hermitian data via NAC divisors of index 1. Notably, even $b_1$ yields generalized Kähler structures with $T = K_S^{-1}$ on surfaces with trivial or ruled canonical bundle; VII$_0$ yields Hopf and Inoue–Bombieri surfaces with Hopf bi-Hermitian examples and Inoue–Bombieri exclusions; VII$_0^+$ forces the surface to be Kato, where LCK and bi-Hermitian theories intertwine yet face constraints from Dloussky data and index obstructions. The work also shows that intermediate Kato surfaces present natural obstructions to LCK-based bi-Hermitian construction, framing open questions about the existence of bi-Hermitian structures in that broad class and connecting the Vaisman problem to the GSS conjecture.

Abstract

We report on a few interrelations between bi-Hermitian metrics and locally conformally Kähler metrics on complex surfaces.

Bi-Hermitian and locally conformally Kähler surfaces

TL;DR

We address the problem of identifying compact complex surfaces that admit bi-Hermitian structures—two integrable, -orthogonal complex structures with the same orientation—while clarifying their relationship with locally conformally Kähler (LCK) geometry. The approach builds the fundamental divisor and the fundamental flat bundle , analyzes Lee forms and Gauduchon degrees, and stratifies surfaces by and Kodaira class; key results connect LCK data to bi-Hermitian data via NAC divisors of index 1. Notably, even yields generalized Kähler structures with on surfaces with trivial or ruled canonical bundle; VII yields Hopf and Inoue–Bombieri surfaces with Hopf bi-Hermitian examples and Inoue–Bombieri exclusions; VII forces the surface to be Kato, where LCK and bi-Hermitian theories intertwine yet face constraints from Dloussky data and index obstructions. The work also shows that intermediate Kato surfaces present natural obstructions to LCK-based bi-Hermitian construction, framing open questions about the existence of bi-Hermitian structures in that broad class and connecting the Vaisman problem to the GSS conjecture.

Abstract

We report on a few interrelations between bi-Hermitian metrics and locally conformally Kähler metrics on complex surfaces.
Paper Structure (12 sections, 10 theorems, 14 equations)

This paper contains 12 sections, 10 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Fundamental divisor and flat bundles
  3. The fundamental line bundle of a bi-Hermitian metric.
  4. Lee bundle of a LCK metric
  5. Gauduchon degree
  6. bi-Hermitian surfaces
  7. Käherian case.
  8. Non-Kählerian surfaces.
  9. Surfaces in class VII$_0$ and $b_2(S)=0$
  10. Surfaces in class VII$_0^+$
  11. Bi-hermitian and LCK metrics, an interplay
  12. Kato surfaces

Key Result

Theorem 1.2

sa94 Each point $p\in (M^4,g,or.)$ of an oriented Riemannian four-manifold belongs to a neighborhood in which there are either zero, one, two or infinitely many complex structures (up to sign) which are $g-$orthogonal, inducing the given orientation.

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.5
  • Proposition 2.1
  • Definition 2.2
  • Corollary 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • ...and 7 more