[Paper Review] Linear foliations of complex spheres I. Chains

Laurent Dufloux|arXiv (Cornell University)|Apr 26, 2017

Geometric Analysis and Curvature Flows3 citations

TL;DR

This paper generalizes the classical Marstrand projection theorem to the complex setting using coordinate-free methods based on Hermitian Grassmann algebra. It establishes dimension-preserving projection results for Borel sets on complex spheres under foliations by complex chains, showing that almost every such foliation preserves the Hausdorff dimension up to the expected bound, extending real Euclidean results to complex geometry with intrinsic geometric structure.

ABSTRACT

We provide coordinate-free versions of the classical projection Theorem of Marstrand-Kaufman-Mattila. This allows us to generalize this Theorem to the complex setting; in restriction to complex spheres, we obtain further projection Theorems along so-called complex chains.

Motivation & Objective

• To extend the classical Marstrand–Kaufman–Mattila projection theorem to the complex setting using coordinate-free geometric methods.
• To define and analyze linear foliations of complex projective spaces via generalized radial projections and Hermitian Grassmann algebra.
• To establish dimension-preserving projection theorems for Borel sets on complex spheres under foliations by complex chains (k-chains).
• To demonstrate that the dimension of a set transverse to a random complex chain foliation on a complex sphere is almost surely min{s, 2n−1−k}, where s is the set's Hausdorff dimension.
• To show that the real case of the result reduces to the classical Marstrand theorem, while the complex case yields a non-trivial strengthening by restricting to a lower-dimensional family of foliations.

Proposed method

• Uses the Hermitian Grassmann (bi)algebra over a complex vector space to provide a coordinate-free framework for geometric analysis.
• Defines generalized radial projections from points in projective space to map sets onto quotient spaces, enabling transversality analysis.
• Endows the codomain of projections with a canonical metric derived from the angular distance in projective space.
• Applies a product formula in the Grassmann algebra to analyze transversality and dimension behavior under projections.
• Applies the theory to complex spheres S^{2n−1} by identifying k-chains as boundaries of totally geodesic submanifolds in complex hyperbolic space.
• Reduces the complex case to affine and spherical foliations via stereographic projection, comparing with the real case.

Experimental results

Research questions

• RQ1 Can the Marstrand projection theorem be generalized to complex projective spaces using intrinsic geometric tools?
• RQ2 What is the transverse Hausdorff dimension of a Borel set on a complex sphere under foliation by complex chains?
• RQ3 How does the dimension of the family of complex chain foliations compare to the full family of real foliations in the same dimension?
• RQ4 To what extent do coordinate-free methods in Grassmann algebra simplify or clarify complex geometric projection theorems?
• RQ5 Does the restriction of foliations to complex chains preserve the almost sure dimension behavior seen in the classical real case?

Key findings

• For a Borel subset A ⊂ S^{2n−1} of Hausdorff dimension s, the transverse dimension with respect to almost every k-chain foliation is min{s, 2n−1−k}.
• The result holds for complex chains, which are special (2k−1)-spheres in S^{2n−1}, arising as boundaries of totally geodesic complex hyperbolic submanifolds.
• In the real case, the result reduces to the classical Marstrand–Kaufman–Mattila theorem, confirming consistency of the framework.
• The space of complex chain foliations has real dimension 2(n−1−k)(k+1), which is half the dimension of the full space of real (2k+2)-planes in R^{2n}, showing a significant restriction.
• The use of Hermitian Grassmann algebra enables a clean, coordinate-free derivation of the projection theorems, especially in the complex setting where coordinate methods would be unwieldy.
• The generalized radial projection construction allows for a unified treatment of foliations in both real and complex projective spaces.