Dolbeault cohomologies of blowing up complex manifolds

One of the fundamental aspects of Complex Geometry is to understand the behavior of various invariants or properties of compact complex manifolds under birational (or bimeromorphic) transformations. By the Weak Factorization Theorem [1], any birational transformation is decomposed into a finite sequence of blow-ups and blow-downs. Hence, to understand the birational invariance of invariants or properties of compact complex manifolds, it suffices to study their behavior under the blow-up transformations.

In this paper, by introducing a notion of relative Dolbeault cohomology, we derive a blow-up formula for Dolbeault cohomology of compact complex manifolds. As applications, we present a uniform viewpoint on the birational invariance of (•,0)- and (0,•)-Hodge numbers on compact complex manifolds and we obtain the birational invariance of the E1-degeneracy of Hodge-to-de Rham spectral sequence for compact complex threefolds and fourfolds.

[1] D. Abramovich, K. Karu, K. Matsuki, J. Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002) 531-572.

https://doi.org/10.1016/j.matpur.2019.01.016