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$\partial\bar{\partial}$-Lemma and $p$-Kähler structures on families of solvmanifolds

Andrea Cattaneo, Adriano Tomassini

Abstract

We provide families of compact $(n + 1)$-dimensional complex non Kähler manifolds satisfying the $\partial\bar{\partial}$-Lemma, with holomoprhically trivial canonical bundle, carrying a balanced metric and with no $p$-Kähler structures. Such a construction extends to the completely solvable case in any dimension Nakamura's construction of low-dimensional holomorphically parallelizable solvmanifolds.

$\partial\bar{\partial}$-Lemma and $p$-Kähler structures on families of solvmanifolds

Abstract

We provide families of compact -dimensional complex non Kähler manifolds satisfying the -Lemma, with holomoprhically trivial canonical bundle, carrying a balanced metric and with no -Kähler structures. Such a construction extends to the completely solvable case in any dimension Nakamura's construction of low-dimensional holomorphically parallelizable solvmanifolds.
Paper Structure (11 sections, 15 theorems, 74 equations)

This paper contains 11 sections, 15 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of Nakamura manifolds
  4. Metric and cohomological properties of Nakamura manifolds
  5. Balanced metrics on N
  6. p-Kählerianity of N
  7. Kodaira dimension
  8. Dolbeault cohomology
  9. de Rham cohomology
  10. The del-delbar-Lemma
  11. Fibred products and explicit examples

Key Result

Theorem 1.1

Let $M \in \operatorname{SL}(n, \mathbb{Z})$ be digonalizabile over $\mathbb{R}$ with positive eigenvalues and let $P \in \operatorname{GL}(n, \mathbb{R})$ be such that $PMP^{-1}$ is diagonal. Let $\tau \in \mathbb{R} \smallsetminus \{ 0 \}$. Then there exists an $(n + 1)$-dimensional compact comple

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • ...and 28 more