Manifolds with parallel differential forms and Kaehler identities for G_2-manifolds

Abstract

Let M be a compact Riemannian manifold equipped with a parallel differential form ω. We prove a version of Kaehler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquotient $(H_c^{\ast }(M),d)$, called {\bf the pseudocohomology} of M. When M is compact and Kaehler and ωis its Kaehler form, $(H_c^{\ast }(M),d)$ is isomorphic to the cohomology algebra of M. This gives another proof of homotopy formality for Kaehler manifolds, originally shown by Deligne, Griffiths, Morgan and Sullivan. We compute $H_c^i(M)$ for a compact G_2-manifold, showing that it is isomorphic to cohomology unless i=3,4. For i=3,4, we compute $H_c^{\ast }(M)$ explicitly in terms of the first order differential operator $*d: Λ^3(M)\arrow Λ^3(M)$.

34 pages, minor corrections, bibliography expanded

References in corpus (3)

• M theory and Singularities of Exceptional Holonomy Manifolds
• Formality of k-connected spaces in 4k+3 and 4k+4 dimensions
• Mirror symmetry aspects for compact G_2 manifolds

Cited by in corpus (11)

• G_2-manifolds and associative submanifolds via semi-Fano 3-folds
• Plurisubharmonic Functions in Calibrated Geometries
• Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds
• The ${\mathcal{L}}_B$-cohomology on compact torsion-free ${\mathrm{G}}_2$ manifolds and an application to 'almost' formality
• Plurisubharmonic functions in calibrated geometry and q-convexity
• Yang-Mills connections on $G_2$-manifolds and Calabi-Yau 3-folds
• $L^2$ harmonic forms on complete special holonomy manifolds
• Compact holonomy ${\mathrm{G}}_2$ manifolds need not be formal
• Vanishing theorems on complete Riemannian manifold with a parallel $1$-form
• Geometry of minimal energy Yang-Mills connections
• Stable Yang-Mills connections on Special Holonomy Manifolds