Complex analogues of the Tverberg--Vre\'cica conjecture and central transversal theorems

TL;DR

This paper develops complex analogues of the Tverberg--Vrécica conjecture and central transversal theorems, extending classical geometric results into the complex setting using equivariant topology.

Contribution

It introduces complex versions of key geometric theorems, proving them via equivariant Euler classes and establishing new Borsuk--Ulam-type results on complex Stiefel manifolds.

Findings

Proved complex Tverberg--Vrécica conjecture and its colorful version.

Established new Borsuk--Ulam-type theorems on complex Stiefel manifolds.

Provided complex analogues of the ham sandwich theorem extensions.

Abstract

The Tverberg--Vre\'cica conjecture claims a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this manuscript, we establish complex analogues of these results, where the corresponding transversals are complex affine spaces. The proofs of the complex Tverberg--Vre\'cica conjecture and its optimal colorful version rely on the non-vanishing of an equivariant Euler class. Furthermore, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments by Axelrod-Freed and Sober\'on, and provide a direct proof of the complex central transversal theorem.