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Bott-Chern formality and Massey products on strong Kähler with torsion and Kähler solvmanifolds

Tommaso Sferruzza, Adriano Tomassini

Abstract

We study the interplay between geometrically-Bott-Chern-formal metrics and SKT metrics. We prove that a $6$-dimensional nilmanifold endowed with a invariant complex structure admits an SKT metric if and only if it is geometrically-Bott-Chern-formal. We also provide some partial results in higher dimensions for nilmanifolds endowed with a class of suitable complex structures. Furthermore, we prove that any Kähler solvmanifold is geometrically formal. Finally, we explicitly construct lattices for a complex solvable Lie group in the list of Nakamura [23] on which we provide a non vanishing quadruple $ABC$-Massey product.

Bott-Chern formality and Massey products on strong Kähler with torsion and Kähler solvmanifolds

Abstract

We study the interplay between geometrically-Bott-Chern-formal metrics and SKT metrics. We prove that a -dimensional nilmanifold endowed with a invariant complex structure admits an SKT metric if and only if it is geometrically-Bott-Chern-formal. We also provide some partial results in higher dimensions for nilmanifolds endowed with a class of suitable complex structures. Furthermore, we prove that any Kähler solvmanifold is geometrically formal. Finally, we explicitly construct lattices for a complex solvable Lie group in the list of Nakamura [23] on which we provide a non vanishing quadruple -Massey product.
Paper Structure (14 sections, 21 theorems, 215 equations)

This paper contains 14 sections, 21 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary
  3. SKT metrics and geometrically-$BC$-formal metrics in complex dimension 3
  4. Family $(I)$
  5. Case $\lambda_1=0$
  6. $\lambda_6=0$
  7. Family $(II)$
  8. Case $\lambda_4\neq 0$
  9. Case $\lambda_4=0$
  10. SKT metrics and Bott-Cher-geometrically formal metrics in higher dimensions
  11. Kähler solvmanifolds and Bott-Chern formality
  12. Quadruple ABC-Massey product
  13. Case: $\mu=\pi$.
  14. Case: $\mu=\frac{\pi}{2}$.

Key Result

Theorem 1

Let $(\Gamma\backslash G:=M,J)$ be a $6$-dimensional nilmanifold endowed with an invariant complex structure $J$. Then, $(M,J)$ is SKT if, and only if, it is geometrically-$BC$-formal. In particular, every Hermitian invariant metric is SKT if, and only if, it is geometrically-$BC$-formal.

Theorems & Definitions (43)

  • Theorem : see Theorem \ref{['thm:SKT-geomBC-dim3']}
  • Theorem : see Theorem \ref{['thm:formality_KS']} and Corollary \ref{['cor:ABC_prod_KS']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 4.1
  • Proposition 4.2
  • ...and 33 more