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Quantization of canonical cones of algebraic curves

B. Enriquez, A. Odesskii

Abstract

We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve C, based on the theory of formal pseudodifferential operators. When C is a complex curve with Poincaré uniformization, we propose another, equivalent construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the quantum algebra when C is a rational curve, and discuss the problem of constructing algebraically "differential liftings".

Quantization of canonical cones of algebraic curves

Abstract

We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve C, based on the theory of formal pseudodifferential operators. When C is a complex curve with Poincaré uniformization, we propose another, equivalent construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the quantum algebra when C is a rational curve, and discuss the problem of constructing algebraically "differential liftings".

Paper Structure

This paper contains 30 sections, 17 theorems, 144 equations.

Table of Contents

  1. Poisson algebras associated with canonical cones of curves
  2. Explicit form of the Poisson ring $A$ for genus $3,4,5$
  3. Quantization based on formal pseudodifferential operators
  4. Outline of the construction
  5. Details of the construction
  6. The algebras $\Psi\operatorname{DO}(R,\partial)$
  7. Functoriality properties of the rings $\Psi\operatorname{DO}(R,\partial)$ and $\Psi\operatorname{DO}(R,\partial)_{\leq 0}$
  8. Construction of $B_X$
  9. Vector field-independent construction of the algebras $B_X^{\operatorname{rat}}$
  10. Proof of Theorem
  11. The case of nonzero divisor $D$
  12. Twisting by generalized line bundles
  13. Functoriality in $(C,D)$
  14. The Poisson algebras
  15. Quantization of the morphisms $\varphi^*_{\operatorname{class}}$
  16. ...and 15 more sections

Key Result

Proposition 1.1

The bracket $\{,\}_\alpha$ is independent on $\alpha$. We denote it by $\{,\}$. It is a Poisson bracket on $A^{\operatorname{rat}}$, taking $A_i^{\operatorname{rat}} \otimes A_j^{\operatorname{rat}}$ to $A_{i+j+1}^{\operatorname{rat}}$. It restricts to a Poisson bracket on $A^{(D)}$. When the effect

Theorems & Definitions (22)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 2.1
  • Remark 1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.1
  • ...and 12 more