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

The Tverberg--Vre\'cica conjecture is 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 proof of the complex Tverberg--Vre\'cica conjecture for powers of the same prime relies on the non-vanishing of an equivariant Euler class. Additionally, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds, which are interesting on their own. 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.