Baker-Sprindzhuk conjectures for complex analytic manifolds

TL;DR

This paper proves that many analytic submanifolds of complex space are strongly extremal, extending classical Diophantine approximation results and solving longstanding conjectures in the complex setting.

Contribution

It generalizes Sprindzhuk's solution to the complex Mahler problem and confirms complex analogues of Baker and Sprindzhuk's conjectures from the 1970s.

Findings

Many analytic submanifolds of C^n are strongly extremal.

The proof uses a variation of quantitative nondivergence estimates.

The results settle complex analogues of classical Diophantine conjectures.

Abstract

We show a large class of analytic submanifolds of C^n to be strongly extremal. This generalizes V. Sprindzhuk's solution of the complex case of Mahler's Problem, and settles complex analogues of conjectures made in the 1970s by Baker and Sprindzhuk. The proof is based on a variation of quantitative nondivergence estimates for quasi-polynomial flows on the space of lattices.