Semi-topological Galois cohomology and Weierstrass realizability

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.

💡 Research Summary

The paper develops a cohomology theory attached to the absolute semi‑topological Galois group Π_ST(X,x), a profinite group obtained as the inverse limit of deck groups of canonical splitting coverings associated to monic Weierstrass polynomials on a Hausdorff, path‑connected space X. For a discrete Π_ST‑module A, the semi‑topological Galois cohomology is defined as Hⁿ_ST(X,A)=Hⁿ_cont(Π_ST(X),A). The authors construct a natural comparison map Φⁿ_X : Hⁿ_ST(X,A) → Hⁿ_sing(X,A) by composing the pull‑back along the canonical surjection ρ : ̂π₁(X) → Π_ST(X) with the classifying map X → Bπ₁(X).

A Lyndon–Hochschild–Serre (LHS) spectral sequence associated to the exact sequence 1 → K_ST(X) → ̂π₁(X) → Π_ST(X) → 1 is established: E₂^{p,q}=H^p_cont(Π_ST(X), H^q_cont(K_ST(X),A)) ⇒ H^{p+q}_cont(̂π₁(X),A). This spectral sequence measures the failure of ρ to be an isomorphism; non‑trivial K_ST yields cohomological obstructions to the surjectivity of Φ.

The paper then studies “semi‑topological embedding problems”. In degree 1, H¹_ST classifies A‑torsors in the semi‑topological Galois category, and a class χ∈H¹_sing(X,A) is realizable by a Weierstrass polynomial precisely when it factors through the deck group of some splitting covering. In degree 2, central extensions 1→A→H→G→1 are classified by H²(G,A); the LHS spectral sequence shows that such an extension lifts to a semi‑topological covering iff its class maps to zero in H²(K_ST,A). Thus the obstruction theory for embedding problems is expressed entirely in terms of the semi‑topological kernel K_ST.

Structural results are proved:

• If π₁(X) is free, then Π_ST(X)≅̂π₁(X) (ST‑fullness), so Φⁿ_X is an isomorphism for all n.
• If π₁(X) is finite, Π_ST(X) is trivial, and all higher semi‑topological cohomology groups vanish.
• For the torus T², K_ST is identified with the subgroup 2ℤ²⊂ℤ², leading to surjectivity of Φ² for all finite coefficients; this argument extends to any complex abelian variety.

The authors introduce the “π₁‑detectable Weierstrass realizability conjecture”: every cohomology class in H²_sing(X,ℤ/m) that is detected by the ordinary fundamental group (i.e., lies in the image of H²(π₁(X),ℤ/m)) should lie in the image of Φ²_X. They verify the conjecture for three broad families:

1. Complex abelian varieties – using the torus computation and the fact that K_ST is a finite‑index subgroup of the lattice, the LHS spectral sequence collapses, giving Φ² surjective.
2. Smooth complex projective curves – their fundamental groups are free (for genus ≥1) or trivial (genus 0), so ST‑fullness applies.
3. Ruled surfaces over curves of positive genus – the fundamental group is a product of a free group with a finite group; the finite factor contributes trivially to Π_ST, while the free factor yields ST‑fullness, again giving surjectivity.

Beyond divisor class realizability, the paper shows that a finite projective representation of Π_ST becomes linear after pulling back to a suitable splitting covering exactly when its associated Schur multiplier class is realized in H²_ST.

In summary, the work provides a systematic cohomological framework for semi‑topological Galois groups, connects it to classical singular cohomology via a natural comparison map, and leverages the resulting spectral sequence to solve embedding problems and verify a broad realizability conjecture for divisor classes on several important classes of complex manifolds. The techniques open avenues for further exploration of higher‑degree obstructions, non‑trivial coefficient systems, and applications to explicit polynomial constructions in complex geometry.