Semi-topological Galois cohomology and Weierstrass realizability

TL;DR

This paper develops a cohomology theory for semi-topological Galois groups associated with Weierstrass polynomials, providing new tools for understanding monodromy and realizability in algebraic geometry.

Contribution

It introduces a cohomology framework for semi-topological Galois groups and applies it to solve problems in monodromy lifting and realizability conjectures.

Findings

Established fundamental properties and comparison maps for the cohomology theory.

Proved structural and vanishing results for semi-topological Galois groups.

Confirmed the $oldsymbol{ ext{pi}_1}$-detectable Weierstrass realizability conjecture for several classes of varieties.

Abstract

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\PiST(X,x)$. This paper develops a cohomology theory for $\PiST(X,x)$ with discrete torsion coefficients, establishing its fundamental properties and canonical comparison maps to singular cohomology. A Lyndon-Hochschild-Serre spectral sequence is used to yield an obstruction theory for semi-topological embedding problems. We prove several structural and vanishing results, including ST-fullness for free fundamental groups and triviality for finite fundamental groups. As applications, we provide a criterion for lifting finite projective monodromy to linear monodromy, formulate the $\pi_1$-detectable Weierstrass realizability conjecture for divisor classes and show that this conjecture is true for abelian varieties, smooth complex projective curves and ruled surfaces over positive-genus curves.