The Double Complex of a Blow-up

Abstract

We compute the double complex of smooth complex-valued differential forms on projective bundles over and blow-ups of compact complex manifolds up to a suitable notion of quasi-isomorphism. This simultaneously yields formulas for 'all' cohomologies naturally associated with this complex (in particular, de-Rham, Dolbeault, Bott-Chern and Aeppli).

Inlcuded an example and discussions of product structures and extensions to singular spaces. Minor improvements throughout. Mainly following referee suggestions

References in corpus (4)

• The cohomologies of the Iwasawa manifold and of its small deformations
• Cohomologies of deformations of solvmanifolds and closedness of some properties
• Mayer-Vietoris systems and their applications
• Morse-Novikov cohomology for blow-ups of complex manifolds

Cited by in corpus (12)

• Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms
• On the Structure of Double Complexes
• Mayer-Vietoris systems and their applications
• Dolbeault and Bott-Chern formalities: deformations and $\partial \overline{\partial }$-lemma
• Hypercohomologies of truncated twisted holomorphic de Rham complexes
• Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology
• On the deformed Bott-Chern cohomology
• Two cohomology theories for structured spaces
• The $\partial \bar{\partial }$-lemma for general Clemens manifolds
• The $\partial \bar{\partial }$-lemma under surjective maps
• Blow-up formulae for twisted cohomologies with supports
• The heredity and bimeromorphic invariance of the $\partial \overline{\partial }$-lemma property