Blow-up formulae for twisted cohomologies with supports

Abstract

We study twisted cohomologies with paracompactifying families of supports. The Künneth theorems, Leray-Hirsch theorems and self-intersection formulae are established. Based on these results, we eventually give explicit expressions of complex blow-up formulae for twisted Dolbeault cohomology on arbitrary complex manifolds and the ones of generalized blow-ups formulae for twisted de Rham cohomology on arbitrary oriented smooth manifolds. These expressions are induced by the morphisms of (simple or double) complexes of spaces of forms and currents rather than just the maps between cohomologies, which help us to obtain the corresponding results for twisted Bott-Chern, Aeppli cohomologies and hypercohomologies of truncated twisted holomorphic de Rham complexes.

Bott-Chern hypercohomology and bimeromorphic invariants

“…Recently, extensive works have been done on the topics of blow-up formulae and the ∂ ∂-lemma. We refer the readers to [6], [7], [14], [42], [28], [29], [30], [31], [32], [51], [53], [47], [23], [22], [34], [38] etc., and the references therein for some resent results.…”

“…Akin to Proposition B.2, we can establish a projective bundle formula for the truncated holomorphic de Rham cohomology considered in Theorem 3.9. Then the remainder of the proof go through by using the same arguments in the proof of Theorem 3.7; see also [29,Theorem 1.4], [31,Theorem 4.18], or [15,Proposition 8].…”

“…Recently, Meng [31] gave a systemical study of twisted cohomologies with supports on complex manifolds, and established blow-up formulae for twisted Dolbeault and Bott-Chern cohomologies.…”