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Sequences of multiple products and cohomology classes for foliations of complex curves

A. Zuevsky

TL;DR

This work develops an algebraic-analytic cohomology theory for codimension-one foliations on complex curves via vertex-algebra–inspired matrix elements. It introduces regularized sequences of multiple products of $ ext{W}$-spaces, along with a rho-sewing–driven geometric model, to produce explicit higher-order foliation invariants. The main technical contributions include convergence results, symmetry and derivative properties, and a canonical, coordinate-invariant framework for spaces of families of complexes, culminating in Godbillon-Vey–type cohomology classes that persist under transversality constraints. This approach links vertex-algebra techniques with classical foliation cohomology, offering new higher-order invariants and potential generalizations to broader geometric settings. The constructions rely on carefully controlled regularization, configuration-space management, and interplays between VOA module data and foliation geometry to obtain robust, nontrivial invariants.

Abstract

The idea of transversality is explored in the construction of cohomology theory associated to regularized sequences of multiple products of rational functions associated to vertex algebra cohomology of codimension one foliations on complex curves. Explicit formulas for cohomology invariants results from consideration transversality conditions applied to sequences of multiple products for elements of chain-cochain transversal complexes defined for codimension one foliations.

Sequences of multiple products and cohomology classes for foliations of complex curves

TL;DR

This work develops an algebraic-analytic cohomology theory for codimension-one foliations on complex curves via vertex-algebra–inspired matrix elements. It introduces regularized sequences of multiple products of -spaces, along with a rho-sewing–driven geometric model, to produce explicit higher-order foliation invariants. The main technical contributions include convergence results, symmetry and derivative properties, and a canonical, coordinate-invariant framework for spaces of families of complexes, culminating in Godbillon-Vey–type cohomology classes that persist under transversality constraints. This approach links vertex-algebra techniques with classical foliation cohomology, offering new higher-order invariants and potential generalizations to broader geometric settings. The constructions rely on carefully controlled regularization, configuration-space management, and interplays between VOA module data and foliation geometry to obtain robust, nontrivial invariants.

Abstract

The idea of transversality is explored in the construction of cohomology theory associated to regularized sequences of multiple products of rational functions associated to vertex algebra cohomology of codimension one foliations on complex curves. Explicit formulas for cohomology invariants results from consideration transversality conditions applied to sequences of multiple products for elements of chain-cochain transversal complexes defined for codimension one foliations.
Paper Structure (28 sections, 20 theorems, 129 equations)

This paper contains 28 sections, 20 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. The main result of the paper
  3. Transversal structures for a foliation
  4. Geometric setup for a foliation in terms of a vertex algebra
  5. The regularization of transversal operators
  6. Sequences of multiple products
  7. The geometric motivation for multiple products of $\mathcal{W}$-spaces
  8. The elimination of coinciding parameters in multiple products
  9. The regularization of multiple product sequences
  10. The product of $\mathcal{W}$-spaces
  11. Convergence of multiple products sequences
  12. Symmetry properties
  13. The existence, $L_V(-1)$-derivative, and $L_V(0)$-conjugation properties
  14. Canonical properties of the $\mathcal{W}$-products
  15. Spaces for families of complexes
  16. ...and 13 more sections

Key Result

Theorem 1

For families of complexes $\left\{ C^{k_i}_{m_i}\left(V, \mathcal{W}^{(i)}, \mathcal{F} \right) \right\}$, $1 \le i \le n$, the sequence of multiple products gendef, the coboundary operators deltaproduct, halfdelta, the transversality condition ortho applied to the families of cochain complexes hat-

Theorems & Definitions (31)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 4
  • proof
  • ...and 21 more