Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class

Lorenzo Sillari, Adriano Tomassini

TL;DR

The paper addresses the Kodaira dimension for almost complex 4-manifolds with torsion $c_1$ by developing a pseudoholomorphic line-bundle framework and proving that the Iitaka dimension—and hence the Kodaira dimension—takes only nonpositive values under tameness. It introduces and analyzes the Iitaka dimension of PH line bundles, showing $κ(M,L,ar∂_L) ∈ {−∞,0}$ and linking $κ_J$ to the PH torsion property of the canonical bundle. It then develops an infinitesimal deformation theory for structures with PH torsion canonical bundle, establishing tangent-space descriptions and unobstructedness akin to Bogomolov–Tian–Todorov, with concrete application to K3 and Enriques surfaces. Together, these results yield a precise local picture of non-integrable deformations with Kodaira dimension zero and illuminate the structure of the moduli of PH-torsion canonical bundle deformations in complex surface settings.

Abstract

In this paper we investigate the Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class. First, we prove that, if the almost complex structure is also tamed, the only possible values for the Kodaira dimension are $0$ or $-\infty$. This is done by developing the theory of pseudoholomorphic structures on vector bundles. In arbitrary dimension, we study infinitesimal deformations of structures with pseudoholomorphically torsion canonical bundle. We compute their tangent space and, under suitable assumptions, we prove an unobstructedness theorem in the spirit of Bogomolov--Tian--Todorov. Together, our results allow to fully describe non-integrable infinitesimal deformations of complex structures on $K3$ and Enriques surfaces in terms of their Kodaira dimension.

Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class

TL;DR

The paper addresses the Kodaira dimension for almost complex 4-manifolds with torsion by developing a pseudoholomorphic line-bundle framework and proving that the Iitaka dimension—and hence the Kodaira dimension—takes only nonpositive values under tameness. It introduces and analyzes the Iitaka dimension of PH line bundles, showing and linking to the PH torsion property of the canonical bundle. It then develops an infinitesimal deformation theory for structures with PH torsion canonical bundle, establishing tangent-space descriptions and unobstructedness akin to Bogomolov–Tian–Todorov, with concrete application to K3 and Enriques surfaces. Together, these results yield a precise local picture of non-integrable deformations with Kodaira dimension zero and illuminate the structure of the moduli of PH-torsion canonical bundle deformations in complex surface settings.

Abstract

In this paper we investigate the Kodaira dimension of almost complex -manifolds with torsion first Chern class. First, we prove that, if the almost complex structure is also tamed, the only possible values for the Kodaira dimension are or . This is done by developing the theory of pseudoholomorphic structures on vector bundles. In arbitrary dimension, we study infinitesimal deformations of structures with pseudoholomorphically torsion canonical bundle. We compute their tangent space and, under suitable assumptions, we prove an unobstructedness theorem in the spirit of Bogomolov--Tian--Todorov. Together, our results allow to fully describe non-integrable infinitesimal deformations of complex structures on and Enriques surfaces in terms of their Kodaira dimension.

Paper Structure

This paper contains 12 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Pseudoholomorphic vector bundles
  3. Preliminaries and notation.
  4. Pseudoholomorphic vector bundles.
  5. Line bundles and their Iitaka dimension.
  6. Iitaka dimension of PH structures of line bundles with
  7. Deformations of structures with PH torsion canonical bundle
  8. Deformation of pairs .
  9. Tangent space to the space of structures with PH trivial canonical bundle.
  10. Tangent space to the space of structures with PH torsion canonical bundle.
  11. Unobstructedness.
  12. Application to complex surfaces.

Key Result

Lemma 2.9

Two PH structures $(E, \bar{\partial}_E)$ and $(F, \bar{\partial}_F)$ are PH equivalent if and only if the corresponding normally integrable PH vector bundles $(E, \mathcal{J}_E)$ and $(F, \mathcal{J}_F)$ are PH equivalent.

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 21 more