Intersection theory and Chern classes in Bott-Chern cohomology

Abstract

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincaré and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.

Chern currents of coherent sheaves

“…Chern classes of coherent sheaves, without the assumption of the existence of a global locally free resolution, were studied in the thesis of Green, [Gre80], as well as in various recent papers, including [Gri10,Hos20a,Hos20b,Qia16,BSW21,Wu20]. Several of these papers also concern classes in finer cohomology theories than de Rham cohomology, as for example (rational or complex) Bott-Chern or Deligne cohomology.…”

Bott-Chern hypercohomology and bimeromorphic invariants

“…real) Bott-Chern cohomology can obtained from the Deligne blow-up formulae and the blow-up formulae for truncated anti-holomorphic de Rham cohomology. The blow-up formulae for integral Bott-Chern cohomology have been proven in [15] and Wu [51], independently. In contrast to the integral and real cases, in the case of G = C, the sheaf complex (3.22) has no natural splitting.…”

“…Bismut [11] presented a Riemann-Roch-Grothendieck theorem taking values in Bott-Chern cohomology which generalizes the classical Riemann-Roch-Grothendieck theorem to complex Hermitian geometry. More recently, for an arbitrary coherent sheaf on a compact complex manifold, Wu [51] construct the Chern classes valued in rational Bott-Chern cohomology and established a Riemann-Roch-Grothendieck formula.…”

“…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.…”