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

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.

• The paper introduces b-complex structures that generalize classical integrability on manifolds to spaces with corners and g-corners.
• The paper employs the b-tangent bundle and Lie algebroid framework to prove an infinite-order Newlander-Nirenberg theorem ensuring local holomorphic coordinates near stratified boundaries.
• The paper establishes a formal equivalence between b-complex structures and standard models, significantly impacting log geometry and analytic methods on singular spaces.

B-Complex Manifolds with Generalized Corners: Generalizations of the Newlander-Nirenberg Theorem

Introduction

This paper (2604.22642) develops a new framework for complex geometry on spaces with boundary and corners, generalizing classical complex manifolds to include manifolds with corners and, more broadly, manifolds with generalized corners (g-corners). The approach is based on the b-tangent bundle ('TX), a Lie algebroid construction, and introduces the notion of b-complex structures. The main technical achievement is an infinite-order formal Newlander-Nirenberg type theorem for such spaces, extending the equivalence between local integrability of complex structures and the existence of holomorphic coordinates to the singular and stratified context of corners and g-corners.

Background: Manifolds with Corners and G-Corners

Manifolds with corners are stratified spaces locally modeled on $[0, \infty)^k \times \mathbb{R}^{m-k}$. They allow a natural compactification of noncompact domains, capturing their asymptotic behavior within a differential-geometric framework. However, classical tangent bundles do not behave appropriately at boundary and corner strata, necessitating the introduction of the b-tangent bundle, pioneered by Melrose, which captures behaviors tangent to strata.

Manifolds with g-corners, as introduced by Joyce, generalize this further by allowing local models governed by weakly toric monoids, accommodating non-simplicial corner structures and arising naturally in moduli problems, log geometry, and mirror symmetry.

B-Complex Structures

The paper defines b-almost complex structures via morphisms $J: 'TX \rightarrow 'TX$ satisfying $J^2 = -\mathrm{id}$. Integrability is characterized via a Nijenhuis tensor defined on b-vector fields. The equivalence with classical notions relies on the decomposition of the complexified b-tangent bundle:

$${'TX} \otimes \mathbb{C} = 'T^{1,0}X \oplus 'T^{0,1}X$$

with corresponding involutive subbundles.

Transversality conditions are imposed at each depth $k$ stratum $S_k(X)$ to ensure compatibility with corner stratification. Holomorphicity is interpreted via b-differential operators and Lie algebroid structures associated with 'TX.

Main Results: Formal Newlander-Nirenberg Theorems

The work extends the classical Newlander-Nirenberg theorem—which states that an integrable almost complex structure admits local holomorphic coordinates—to the context of b-complex manifolds with corners and g-corners.

Theorem 3.6 (Local Standard Model at Corners)

For a b-complex manifold with corners or g-corners, local coordinates can be constructed near any point in a depth $k$ stratum such that:

• The b-complex structure $J$ and the associated Lie algebroid structure are put into standard form.
• There exist coordinates $$(\theta_1, \ldots, \theta_k, z_{k+1}, \ldots, z_n) \in \mathbb{R}^k \times \mathbb{C}^{n-k}$$ with $z_j$ holomorphic to infinite order along the stratum.
• Commutativity and flatness conditions on b-normal bundle sections are enforced, guaranteeing compatibility with corner stratification and Lie bracket relations.

Theorem 3.7 (Formal Completion Agreement)

The formal completion of a b-complex structure along each corner stratum agrees to infinite order with a standard model b-complex structure. Explicitly, for any depth $J: 'TX \rightarrow 'TX$0 stratum, the b-complex structure can be formally modeled by a product

$J: 'TX \rightarrow 'TX$1

where $J: 'TX \rightarrow 'TX$2 is a toric monoid associated with the stratum. Holomorphic functions are constructed as monoid algebra elements of the form $J: 'TX \rightarrow 'TX$3, confirming the infinite-order agreement.

Holomorphic Functions at Corners

Standard holomorphic functions in the b-complex setting are degenerate at corners: e.g., $J: 'TX \rightarrow 'TX$4 fails to recover $J: 'TX \rightarrow 'TX$5 when $J: 'TX \rightarrow 'TX$6, highlighting the intrinsic singular behavior. The construction ensures that all formal b-holomorphic functions at the corner are local functions of these canonical elements.

Implications for Log Geometry and Kato-Nakayama Spaces

In the sequel [1], these constructions are utilized to show that Kato-Nakayama spaces associated to log smooth log $J: 'TX \rightarrow 'TX$7-schemes admit b-complex manifold structures with g-corners. Necessary and sufficient conditions for a b-complex manifold with g-corners to arise as a Kato-Nakayama space are derived using the generalized Newlander-Nirenberg theorem. This has substantial implications for mirror symmetry and log-geometric compactifications, as it clarifies the analytic structure underlying log schemes and their associated topological spaces.

Theoretical and Practical Significance

This framework advances the theory of complex structures on singular spaces, providing a robust infinite-order formalism compatible with stratified boundary structures and general singularities. The adoption of Lie algebroids and b-tangent bundles enables the extension of complex analytic methods to settings previously inaccessible via classical geometry.

Practically, the results establish foundations for complex analytic, cohomological, and index-theoretic techniques in moduli problems, log-geometry, and other settings involving natural compactifications with corners and non-simplicial boundary behaviors. Future developments will likely further generalize analytic and geometric invariants, extend Hodge-theoretic constructions, and refine intersection theory on spaces with g-corners.

Conclusion

The paper rigorously extends complex geometry to manifolds with corners and generalized corners, introducing b-complex structures and establishing infinite-order Newlander-Nirenberg type theorems. These results provide a local and formal classification of b-complex structures at each stratum, underpinning analytic constructions in log geometry and beyond. The framework is poised to influence a broad range of geometric and analytic theories involving stratified spaces and singularities.