On the field of meromorphic functions on a Stein surface

TL;DR

This paper establishes key properties of fields of meromorphic functions on Stein surfaces, including cohomological dimension, solutions to classical problems, and extensions to real surfaces with involutions, advancing complex geometry and algebraic function theory.

Contribution

It proves that these fields have cohomological dimension 2 and solves the period-index problem and Serre's conjecture II for them, also addressing real cases with involutions and Hilbert's 17th problem.

Findings

Fields of meromorphic functions on Stein surfaces have cohomological dimension 2.

Solved the period-index problem and Serre's conjecture II for these fields.

Provided an optimal quantitative solution to Hilbert's 17th problem on analytic surfaces.

Abstract

We prove that fields of meromorphic functions on Stein surfaces have cohomological dimension 2, and solve the period-index problem and Serre's conjecture II for these fields. We obtain analogous results for fields of real meromorphic functions on Stein surfaces equipped with an antiholomorphic involution. We deduce an optimal quantitative solution to Hilbert's 17th problem on analytic surfaces.