On the projective structures given by theta

TL;DR

The paper investigates a canonical Theta-based projective structure on a compact genus $g$ Riemann surface $C$, arising from the line in $H^0(J_C, 2\Theta)$ orthogonal to sections vanishing at the origin, and compares it to the uniformization and Hodge-theoretic structures. It proves the key identity $\overline{\partial} \beta^\theta = 8\pi j^* \Theta^* \omega_{FS}$ and shows that this Theta-derived structure generically differs from the Hodge-theoretic one for all genera, while the pullback metric $\Theta^* \omega_{FS}$ descends to $\mathsf{A}_g$. Through degeneration arguments, including the elliptic case, it is established that the Theta-based and Hodge-based projective structures are distinct universally across genus, highlighting a rich interaction between theta data, moduli, and canonical geometric structures on curves. These results provide a global invariant on the moduli of abelian varieties via the descended $(1,1)$ form and clarify how theta-constructs distinguish between different canonical projective structures on curves across genus.

Abstract

Given a compact Riemann surface $C$, the line in $H^0(J_C,\, 2Θ)$ orthogonal to the sections vanishing at $0$ produces a natural projective structure on $C$. We investigate the properties of this projective structure.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• proof : Proof of the Claim