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Bidifferentials, Lagrangian projections and the Virasoro extension

Eduard Looijenga

TL;DR

The paper establishes a deep link between Lagrangian complements of $H^0(C,\Omega_C)$ in $H^1_{dR}(C)$ and symmetric bidifferentials on $C^2$ with biresidue $1$, identifying a natural affine/torsor correspondence via a local residue framework. It then identifies the canonical anti-invariant 2-form $\zeta_C$ (BCFP) with a pure Hodge type cohomology class on $C^2$, and analyzes its action on $H^\bullet(C)$, yielding a semisimple endomorphism aligned with the Hodge decomposition. The work generalizes to higher powers $C^n$ through the forms $\zeta_n$, producing cohomology classes on configuration spaces with explicit geometric and representation-theoretic content, potentially relevant to WZW models. Finally, the authors construct a canonical Virasoro extension of the derivations of a local field via a DVR-residue framework and a Fock-space realization, providing a natural, η-independent description of the Virasoro algebra arising in this geometric setting.

Abstract

Let $C$ be a smooth projective curve over an algebraically closed field $k$ of characteristic zero. We prove that a Lagrangian supplement of $H^0(C, Ω_C)$ in the de Rham cohomology group $H^1_{dR}(C)$ determines and is determined by a particular type of symmetric bidifferential on $C^2$ (its polar divisor must be twice the diagonal and have biresidue one along it). When $k$ is the complex field, a natural choice of such supplement is $H^{0,1}(C)$ and we show that this corresponds with the bidifferential that after a twist is the rational $2$-form on $C^2$ found by Biswas-Colombo-Frediani-Pirola. We determine the cohomology class carried by that $2$-form and define an analogue of this form as rational $n$-form on $C^n$ that is regular on the $n$-point configuration space of $C$. The proof relies on a local version of the above correspondence, which can be stated in terms of a complete discrete valuation ring. We use this local version also to construct in a natural manner the Virasoro extension of the Lie algebra of derivations of a local field.

Bidifferentials, Lagrangian projections and the Virasoro extension

TL;DR

The paper establishes a deep link between Lagrangian complements of in and symmetric bidifferentials on with biresidue , identifying a natural affine/torsor correspondence via a local residue framework. It then identifies the canonical anti-invariant 2-form (BCFP) with a pure Hodge type cohomology class on , and analyzes its action on , yielding a semisimple endomorphism aligned with the Hodge decomposition. The work generalizes to higher powers through the forms , producing cohomology classes on configuration spaces with explicit geometric and representation-theoretic content, potentially relevant to WZW models. Finally, the authors construct a canonical Virasoro extension of the derivations of a local field via a DVR-residue framework and a Fock-space realization, providing a natural, η-independent description of the Virasoro algebra arising in this geometric setting.

Abstract

Let be a smooth projective curve over an algebraically closed field of characteristic zero. We prove that a Lagrangian supplement of in the de Rham cohomology group determines and is determined by a particular type of symmetric bidifferential on (its polar divisor must be twice the diagonal and have biresidue one along it). When is the complex field, a natural choice of such supplement is and we show that this corresponds with the bidifferential that after a twist is the rational -form on found by Biswas-Colombo-Frediani-Pirola. We determine the cohomology class carried by that -form and define an analogue of this form as rational -form on that is regular on the -point configuration space of . The proof relies on a local version of the above correspondence, which can be stated in terms of a complete discrete valuation ring. We use this local version also to construct in a natural manner the Virasoro extension of the Lie algebra of derivations of a local field.

Paper Structure

This paper contains 15 sections, 12 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Bidifferentials and Lagrangian projections
  3. Biresidues and projective structures
  4. The sheaf of projective structures
  5. Brief review of the Rham cohomology of $C$
  6. Lagrangian projections in $H^1_{dR}(C)$
  7. The class of the canonical 2-form
  8. Betti and de Rham cohomology
  9. The canonical $2$-form
  10. A generalization for higher powers of the curve
  11. The form $\zeta_n$
  12. Canonical construction of the Virasoro algebra
  13. Review of the basic Fock representation
  14. The Fock space for a DVR
  15. A canonical description of the Virasoro algebra

Key Result

Proposition 2.3

Let $\eta\in (\hat{\Omega}^{(2)})^\sigma$ have biresidue constant $1$. Then $\operatorname{Res}_{1\to 2}\pi_1^*(f)\eta=df$ for every $f\in K$. Furthermore, the map is zero on ${\mathcal{O}}$ and its image is a Lagrangian supplement of $\Omega$ in $\Omega'_K$. If we use the universal derivation to identify $K'$ with $\Omega'_K$, so that $S_\eta$ factors though an endomorphism $S'_\eta$ of $K'$, th

Theorems & Definitions (30)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.6
  • proof
  • Theorem 3.1
  • Remark 3.2
  • ...and 20 more