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Holomorphic $p$-Contact and $s$-Symplectic Line Bundles

Kyle Broder, Dan Popovici

TL;DR

The paper develops a bundle-valued generalisation of holomorphic $p$-contact and $s$-symplectic structures on compact complex manifolds, showing these structures can exist on projective varieties and are tightly linked to spin geometry via a square root of the anticanonical bundle. It establishes curvature and positivity constraints using singular Hermitian metrics and recent $m$-positivity regularisation, proving that the presence of such structures forces the canonical bundle to be non-pseudo-effective (in many cases implying negative Kodaira dimension and rational connectivity). It also develops the associated sheaves ${ m F}_ Gamma$ and ${ m G}_{ Gamma,h}$, and provides concrete examples in complex projective spaces and their hypersurfaces, illustrating when bundle-valued $p$-contact structures can exist. Overall, the work uncovers deep links between higher-degree holomorphic structures, spin geometry, and positivity properties in complex geometry.

Abstract

We generalise the notions of scalar-valued holomorphic $p$-contact and $s$-symplectic structures introduced recently on compact complex manifolds by the second-named author jointly with H. Kasuya and L. Ugarte to their analogues with values in a holomorphic line bundle. We then study the resulting holomorphic $p$-contact and $s$-symplectic manifolds which, unlike their scalar counterparts that are never Kähler, can even be projective. In particular, we investigate the (lack of) positivity properties of the canonical bundle of these manifolds when it is given a possibly singular Hermitian fibre metric. One of the tools used is a very recent regularisation result for $m$-psh functions obtained jointly by S. Dinew and the second-named author.

Holomorphic $p$-Contact and $s$-Symplectic Line Bundles

TL;DR

The paper develops a bundle-valued generalisation of holomorphic -contact and -symplectic structures on compact complex manifolds, showing these structures can exist on projective varieties and are tightly linked to spin geometry via a square root of the anticanonical bundle. It establishes curvature and positivity constraints using singular Hermitian metrics and recent -positivity regularisation, proving that the presence of such structures forces the canonical bundle to be non-pseudo-effective (in many cases implying negative Kodaira dimension and rational connectivity). It also develops the associated sheaves and , and provides concrete examples in complex projective spaces and their hypersurfaces, illustrating when bundle-valued -contact structures can exist. Overall, the work uncovers deep links between higher-degree holomorphic structures, spin geometry, and positivity properties in complex geometry.

Abstract

We generalise the notions of scalar-valued holomorphic -contact and -symplectic structures introduced recently on compact complex manifolds by the second-named author jointly with H. Kasuya and L. Ugarte to their analogues with values in a holomorphic line bundle. We then study the resulting holomorphic -contact and -symplectic manifolds which, unlike their scalar counterparts that are never Kähler, can even be projective. In particular, we investigate the (lack of) positivity properties of the canonical bundle of these manifolds when it is given a possibly singular Hermitian fibre metric. One of the tools used is a very recent regularisation result for -psh functions obtained jointly by S. Dinew and the second-named author.

Paper Structure

This paper contains 18 sections, 19 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. The $p$-contact setting
  3. Preliminaries
  4. Local holomorphic $p$-contact structures
  5. Links with contact and spin manifolds
  6. Space of all $p$-contact structures
  7. Local holomorphic $p$-no-contact structures
  8. Positivity and curvature
  9. Laplacians induced by singular fibre metrics
  10. Regularising $f$
  11. Curvature of the $p$-contact and canonical bundles
  12. Curvature of the contact and canonical bundles
  13. Scalar curvature of spin manifolds
  14. Local holomorphic $s$-symplectic structures
  15. The sheaves ${\cal F}_\Gamma$ and ${\cal G}_{\Gamma,\,h}$ associated with a holomorphic $p$-contact bundle
  16. ...and 3 more sections

Key Result

Theorem 1.4

Let $(X,\,\omega)$ be a compact Kähler manifold with $\hbox{dim}_\mathbb{C} X = n$. (i) Suppose there exists a spin structure on $X$ and the anti-canonical bundle $-K_X$ possesses a $C^\infty$ Hermitian fibre metric $h^2$ such that Then $H^0(X,\,K_X) = H^{n,\,0}(X,\,\mathbb{C}) = \{0\}$. (ii) Suppose that $n = 2p+1$ and there exist a holomorphic line bundle $L$ on $X$ and an $L$-valued holomo

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.8
  • ...and 16 more