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

This paper, "B-complex manifolds with generalized corners. I. Newlander–Nirenberg Theorems" by Hülya Argüz and Dominic Joyce, presents a significant extension of complex geometry to singular spaces. It bridges the gap between classical complex manifold theory and the geometry of manifolds with corners and "generalized corners" (g-corners).

Below is a detailed technical summary of the work.

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1. The Problem: Extending Integrability to Singular Settings

In classical differential geometry, the Newlander–Nirenberg Theorem provides the fundamental link between almost complex structures and complex manifolds: an almost complex structure $J$ is integrable (arising from holomorphic coordinate charts) if and only if its Nijenhuis tensor $N_J$ vanishes.

The authors identify a limitation in this classical framework: it is designed for smooth manifolds modeled on $\mathbb{R}^m$. However, many modern geometric objects—such as manifolds with corners (modeled on $[0, \infty)^k \times \mathbb{R}^{m-k}$) and manifolds with g-corners (modeled on spaces $X_P$ derived from weakly toric monoids)—possess boundary strata and singular structures that the standard tangent bundle $TX$ cannot handle gracefully. To study these, one must use the b-tangent bundle ($bTX$), which allows for controlled behavior near the boundaries. The central problem is to formulate and prove a version of the Newlander–Nirenberg theorem that accounts for these $b$-structures and the stratified nature of g-corners.

2. Methodology: $b$-Geometry and Monoid Models

The authors employ several advanced mathematical frameworks to tackle the problem:

• Manifolds with g-corners: They utilize the theory of manifolds with generalized corners, where local models are determined by weakly toric monoids $P$. These spaces are more flexible than standard manifolds with corners and are essential for describing the local structure of Kato–Nakayama spaces in log geometry.
• $b$-Tangent Bundles ($bTX$): Instead of the ordinary tangent bundle, the authors work with the $b$-tangent bundle, which is a Lie algebroid. In local coordinates $(x_1, \dots, x_k, x_{k+1}, \dots, x_m)$ where $x_i \ge 0$ for $i \le k$, the $b$-tangent bundle is spanned by $\{x_1\frac{\partial}{\partial x_1}, \dots, x_k\frac{\partial}{\partial x_k}, \frac{\partial}{\partial x_{k+1}}, \dots, \frac{\partial}{\partial x_m}\}$.
• $b$-Complex Structures: A $b$-almost complex structure $J$ is defined as a bundle morphism on $bTX$ such that $J^2 = -\text{id}$. Crucially, they impose a transversality condition on the $b$-normal bundles of the depth strata to ensure the structure is compatible with the corner geometry.
• Formal Completion and Infinite Order Agreement: Because the coordinates at the corners are inherently degenerate (e.g., $\lambda e^{i\theta}$ cannot recover $\theta$ when $\lambda=0$), the authors shift the goal from finding exact holomorphic coordinates to finding coordinates that agree with a "standard model" to infinite order along the corner strata.

3. Key Contributions and Results

A. Definition of $b$-Complex Manifolds with g-corners

The paper establishes a rigorous definition of a $b$-complex manifold. This includes the definition of the Nijenhuis tensor $N_J$ using $b$-vector fields and the characterization of integrability via the $b$-$\bar{\partial}$ operator.

B. Theorem 3.6: Local Structure on Strata

The authors prove that on any depth stratum $S_k(X)$, the $b$-complex structure $J$ behaves predictably. They show that one can find local coordinates that put $J$ into a "standard form" relative to the Lie algebroid structure of $bTX|_{S_k(X)}$. This ensures that the complex structure is "well-behaved" as one approaches the corners.

C. Theorem 3.7: The Formal Newlander–Nirenberg Theorem

This is the paper's primary result. It states that if $(X, J)$ is a $b$-complex manifold with (g-)corners, then near any point $x$ in a stratum $S_k(X)$, there exist local coordinates such that $J$ matches the standard $b$-complex structure ($J_{st}$) to infinite order.

• The "standard" holomorphic functions in these coordinates take the form $(\lambda_1 e^{i\theta_1}, \dots, \lambda_k e^{i\theta_k}, z_{k+1}, \dots, z_n)$.
• This proves that while $J$ might not be globally equivalent to a standard model, it is formally integrable at the boundary.

4. Significance and Applications

The significance of this work is two-fold:

1. Theoretical Foundation: It provides the necessary differential-geometric machinery to treat singular spaces (like those in log geometry) as if they were complex manifolds, provided one uses the $b$-calculus.
2. Log Geometry and Mirror Symmetry: The authors explicitly mention that this theory is designed to characterize Kato–Nakayama spaces ($S_{KN}$). In the sequel (Part II), they show that if $S$ is a log smooth log $\mathbb{C}$-scheme, its Kato–Nakayama space is a $b$-complex manifold with g-corners. This provides a bridge between algebraic log geometry and the differential geometry required for studying Mirror Symmetry.

Summary Table

Feature	Classical Complex Manifold	$b$-Complex Manifold (this paper)
Local Model	$\mathbb{R}^{2n}$	$X_P \times \text{Hom}(P^{gp}, \mathbb{R})$
Tangent Bundle	$TX$|$bTX$ (Lie algebroid)
Integrability	$N_J = 0 \implies$ Holomorphic charts	$N_J = 0 \implies$ Formal holomorphic charts
Coordinate Type	$z_j = x_j + iy_j$	$\lambda_j e^{i\theta_j}$ and $z_j$