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Semi-topological Galois cohomology and Weierstrass realizability

Jyh-Haur Teh

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 $π_1$-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.

Semi-topological Galois cohomology and Weierstrass realizability

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, . This paper develops a cohomology theory for 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 -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.

Paper Structure

This paper contains 25 sections, 27 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The semi-topological Galois category and $\Pi^{\mathrm{ST}}(X,x)$
  3. Weierstrass polynomials and splitting coverings
  4. The indexing system of splitting coverings
  5. Definition of the absolute semi-topological Galois group
  6. Galois category formulation
  7. Continuous cochains and continuous cohomology
  8. Definition of $H_{\mathrm{ST}}^\bullet(X,A)$
  9. Properties
  10. Comparison with classical topology
  11. A comparison map to singular cohomology
  12. Semi-topological embedding problems
  13. A realizability criterion
  14. Degree $2$ and embedding problems
  15. Computations and structural results
  16. ...and 10 more sections

Key Result

Theorem 2.4

For any Weierstrass polynomial $f$ on $X$:

Theorems & Definitions (70)

  • Definition 2.1: Weierstrass polynomial
  • Definition 2.2: Solution space and splitting in a covering
  • Definition 2.3: Splitting covering
  • Theorem 2.4: Existence, uniqueness, and Galois property
  • Definition 2.5: Deck group
  • Lemma 2.6
  • proof
  • Definition 2.7: Absolute semi-topological Galois group
  • Remark 2.8: Basepoint dependence
  • Theorem 2.9
  • ...and 60 more