[Paper Review] Projective structures on a Riemann surface

Indranil Biswas, A.K. Raina|ArXiv.org|Jul 24, 1996

Geometry and complex manifolds5 references4 citations

TL;DR

This paper establishes a canonical identification between projective structures on a compact Riemann surface $X$ of genus $g$ and trivialisations of a specific line bundle $L = K_{X imes X} igotimes O_{X imes X}(2 riangle)$ over the third infinitesimal neighbourhood $3\triangle$, subject to compatibility with a canonical trivialisation over $2\triangle$. The key contribution is a geometric characterization of projective structures via higher-order jet data on the diagonal, linking algebraic geometry to conformal field theory through the Sugawara form.

ABSTRACT

For a compact Riemann surface $X$ of any genus $g$, let $L$denote the line bundle $K_{X imes X}\otimes {\cal O}_{X imes X}(2Δ)$ on $X imes X$, where $K_{X imes X}$ is the canonical bundle of $X imes X$ and $Δ$ is the diagonal divisor. We show that $L$ has a canonical trivialisation over the nonreduced divisor $2Δ$. Our main result is that the space of projective structures on $X$ is canonically identified with the space of all trivialisations of $L$ over $3Δ$ which restrict to the canonical trivialisation of $L$ over $2Δ$ mentioned above. We give a direct identification of this definition of a projective structure with a definition of Deligne.We also describe briefly the origin of this work in the study of the so-called "Sugawara form" of the energy-momentum tensor in a conformal quantum field theory.

Motivation & Objective

• To provide a geometric characterization of projective structures on a compact Riemann surface using higher-order jet data.
• To establish a canonical identification between projective structures and trivialisations of a line bundle $L$ over $3\triangle$ compatible with a trivialisation over $2\triangle$.
• To connect this geometric construction to the Sugawara form in conformal quantum field theory.
• To formalize the space of projective structures as a moduli problem in algebraic geometry via jet-theoretic data.

Proposed method

• Define the line bundle $L = K_{X imes X} \otimes \mathcal{O}_{X imes X}(2\Delta)$ on $X \times X$, where $K_{X\times X}$ is the canonical bundle and $\Delta$ is the diagonal divisor.
• Construct a canonical trivialisation of $L$ over the nonreduced divisor $2\Delta$ using jet-theoretic techniques.
• Characterize projective structures as trivialisations of $L$ over $3\Delta$ that restrict to the canonical trivialisation over $2\Delta$.
• Use the theory of infinitesimal neighbourhoods and line bundle trivialisations to encode projective structures algebraically.
• Establish an equivalence between this geometric definition and the standard definition of projective structures via projective connections.
• Relate the construction to the Sugawara form in conformal field theory through the geometry of the diagonal and jet bundles.

Experimental results

Research questions

• RQ1 How can projective structures on a Riemann surface be canonically characterised using higher-order jet data on the diagonal?
• RQ2 What is the role of the line bundle $L = K_{X\times X} \otimes \mathcal{O}_{X\times X}(2\Delta)$ in encoding projective structures?
• RQ3 How does the canonical trivialisation of $L$ over $2\Delta$ serve as a natural base for classifying projective structures?
• RQ4 In what way does this geometric construction relate to the Sugawara form in conformal field theory?
• RQ5 Can the space of projective structures be fully described as a moduli space of trivialisations over $3\Delta$ compatible with a fixed trivialisation over $2\Delta$?

Key findings

• The space of projective structures on a compact Riemann surface $X$ is canonically identified with the space of trivialisations of the line bundle $L = K_{X\times X} \otimes \mathcal{O}_{X\times X}(2\Delta)$ over $3\Delta$ that restrict to the canonical trivialisation over $2\Delta$.
• A canonical trivialisation of $L$ over $2\Delta$ is constructed using jet-theoretic methods on the diagonal divisor.
• This identification provides a new, geometric definition of projective structures that is intrinsic to the Riemann surface and its product structure.
• The construction establishes a direct equivalence between this jet-theoretic definition and the classical definition via projective connections.
• The work provides a geometric interpretation of the Sugawara form in conformal field theory through the geometry of $L$ and its trivialisations over infinitesimal neighbourhoods of the diagonal.
• The result is valid for any compact Riemann surface of genus $g$, regardless of the value of $g$, showing the universality of the construction.