Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homotopy BV-algebras in Hermitian geometry

Joana Cirici, Scott O. Wilson

Abstract

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative $BV_\infty$-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

Homotopy BV-algebras in Hermitian geometry

Abstract

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative -algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.
Paper Structure (14 sections, 21 theorems, 122 equations)

This paper contains 14 sections, 21 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries
  3. Multicomplexes and the gauge equation
  4. Batalin-Vilkovisky and hypercommutative algebras
  5. Hermitian manifolds
  6. Almost Hermitian and almost symplectic manifolds
  7. Almost Hermitian manifolds
  8. Almost symplectic and almost Kähler manifolds
  9. Symplectic manifolds with real polarizations
  10. Chevalley-Eilenberg algebras and examples
  11. Canonical deformation retracts
  12. Poisson structures
  13. Almost Hermitian structures
  14. Symplectic-Lagrangian structures

Key Result

Proposition 2.4

Let $(A,d,\Delta_1,\Delta_2,\ldots)$ be a multicomplex satisfying the degeneration property and let $(\iota,\rho,h)$ be a deformation retract. A solution to the gauge equation is given by

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof
  • Corollary 2.8
  • Proposition 2.9
  • ...and 30 more