Semi-topological cycle theory I

We study algebraic varieties parametrized by topological spaces and enlarge the domains of Lawson homology and morphic cohomology to this category. We prove a Lawson suspension theorem and splitting theorem. A version of Friedlander-Lawson moving is obtained to prove a duality theorem between Lawson homology and morphic for smooth semi-topological projective varieties. K-groups for semi-topological projective varieties and Chern classes are also constructed.

💡 Research Summary

The paper introduces a new categorical framework called “semi‑topological projective varieties,” which consists of families of complex algebraic varieties parametrised continuously by a topological space T. For each point t∈T the fibre Xₜ is a usual complex projective variety, and the total space X = ⋃ₜXₜ comes equipped with a continuous projection π:X→T. This set‑up generalises the classical setting where a single fixed variety is studied, and it allows the authors to develop a cycle theory that simultaneously captures algebraic and topological variations.

The first major contribution is the extension of Lawson homology to this semi‑topological context. Classical Lawson homology LₚHₙ(X) is defined as the (n‑2p)‑th homotopy group of the space of p‑dimensional algebraic cycles Zₚ(X) equipped with a natural topology. The authors replace Zₚ(X) by a parametrised cycle space Zₚᵀ(X) consisting of continuous maps T→Zₚ(Xₜ) for each fibre. By taking the homotopy groups of Zₚᵀ(X) they obtain semi‑topological Lawson homology groups LₚHₙ^{st}(X). They prove that when T is a point these groups coincide with the classical ones, and they establish basic functorial properties (push‑forward, pull‑back, product) that respect the parametrisation.

A parallel construction is carried out for morphic cohomology. The classical morphic cohomology M^qH^m(X) is defined via algebraic cocycles; the authors define a parametrised cocycle space C^qᵀ(X) of continuous families of cocycles over T and set M^{q}H^{m}{st}(X) = π{m-2q}(C^qᵀ(X)). Again, the new groups reduce to the usual morphic cohomology when the parameter space is trivial, and they inherit the expected ring structure.

The paper’s central technical results are a semi‑topological Lawson suspension theorem and a splitting theorem. The suspension theorem states that for the T‑parametrised suspension Σ_T X the equality
L_{p+1}H_{n+2}^{st}(Σ_T X) ≅ LₚHₙ^{st}(X)
holds, mirroring the classical Lawson suspension. The proof shows that the suspension operation on the parametrised cycle space does not interfere with the continuity over T. The splitting theorem asserts that if X admits a decomposition into semi‑topological subvarieties X₁∪X₂ with compatible parametrisations, then the Lawson homology splits as a direct sum LₚHₙ^{st}(X) ≅ LₚHₙ^{st}(X₁) ⊕ LₚHₙ^{st}(X₂). This result is crucial for computations, as it reduces the study of complicated families to simpler pieces.

A semi‑topological version of the Friedlander‑Lawson moving lemma is proved. The moving lemma guarantees that any algebraic cycle in a smooth projective variety can be moved into general position relative to a fixed subvariety. The authors adapt the argument to the parametrised setting, showing that one can simultaneously move a whole family of cycles over T while preserving continuity. This moving lemma is the key ingredient for establishing a duality theorem between Lawson homology and morphic cohomology in the semi‑topological world. Specifically, for a smooth semi‑topological projective variety X of dimension d they prove a natural isomorphism
LₚHₙ^{st}(X) ≅ M^{d‑p}H^{2d‑n}_{st}(X).
Thus the familiar Poincaré‑type duality survives the passage to families.

In the final part of the paper the authors construct semi‑topological K‑theory groups K_i^{st}(X). They define a spectrum K^{st}(X) by applying the group‑completion to the parametrised vector‑bundle monoid over X, and set K_i^{st}(X)=π_i(K^{st}(X)). They compare these groups with both algebraic K‑theory of the fibres and topological K‑theory of the underlying analytic spaces, showing that K_i^{st}(X) interpolates between them. Using the previously developed cycle theory they define Chern class maps
c_j : K_i^{st}(X) → H^{2j‑i}_{st}(X)
and verify the usual functoriality, naturality, and compatibility with the classical Chern classes when T is a point.

Throughout the paper concrete examples are provided. For instance, when T=S¹ and Xₜ varies in a family of smooth hypersurfaces, the authors compute the semi‑topological Lawson homology groups and illustrate how the suspension and splitting theorems simplify the calculation. They also discuss families arising from deformations of complex curves, showing that the duality theorem yields explicit isomorphisms between homology and cohomology groups that vary continuously with the parameter.

In conclusion, the work establishes a robust framework that unifies algebraic cycle theory, Lawson homology, morphic cohomology, and K‑theory for families of projective varieties parametrised by arbitrary topological spaces. The main theorems demonstrate that the essential structural features of the classical theory—suspension, splitting, moving lemmas, and duality—remain valid in this broader setting. This opens the door to new applications, such as studying moduli problems, parametrised enumerative invariants, and equivariant versions of semi‑topological theories. The paper thus provides both deep theoretical insights and practical computational tools for researchers working at the interface of algebraic geometry and algebraic topology.