B-complex manifolds with generalized corners. I. Newlander-Nirenberg Theorems

TL;DR

This paper extends the concept of complex manifolds to manifolds with corners and generalized corners using b-structures, proving a Newlander-Nirenberg type theorem for these structures.

Contribution

It introduces a formal Newlander-Nirenberg theorem for manifolds with corners and generalized corners, linking complex structures on b-tangent bundles to standard models.

Findings

Proves a Newlander-Nirenberg type theorem for b-complex structures on manifolds with corners.

Shows that Kato-Nakayama spaces of log schemes have b-complex manifold structures.

Provides conditions under which b-complex manifolds with g-corners are Kato-Nakayama spaces.

Abstract

We generalize complex manifolds to manifolds with corners $X$, and to manifolds with generalized corners (g-corners) in the sense of the second author arXiv:1501.00401, using complex structures on the b-tangent bundle (log tangent bundle) ${}^bTX$. We prove a formal Newlander-Nirenberg type theorem showing that along each corner stratum of $X$, the b-complex structure agrees with a standard model to infinite order. In the sequel we show that if $S$ is a log smooth log $\mathbb C$-scheme, or log smooth log complex analytic space, then the Kato-Nakayama space $S^{\rm KN}$ has the structure of a b-complex manifold with g-corners. Using our Newlander-Nirenberg theorem we give necessary and sufficient conditions for a b-complex manifold with g-corners to be a Kato-Nakayama space.