Fourier-Mukai transform for fine compactified Prym varieties

Abstract

Consider a finite covering $β: C \to X$ of a smooth projective curve $X$ by a reduced, projective, planar curve $C$. Associated to two general polarizations on $C$, $q$ and $q'$, one can construct the corresponding compactified Prym varieties $\overline{\mathrm{P}}_β(q)$ and $\overline{\mathrm{P}}_β(q')$. Consider $Γ$ to be the group of line bundles whose torsion coincides with the order of $β$. In this article we construct a Fourier-Mukai transform between the derived categories of $\overline{\mathrm{P}}_β(q)$ and the $Γ$-equivariant derived category of $\overline{\mathrm{P}}_β(q')$. Hence, we obtain a derived equivalence between the $\mathrm{SL}(n,\mathbb{C})$-Hitchin fibre and its associated $\mathrm{PGL}(n,\mathbb{C})$-Hitchin fibre for a dense class of singular spectral curves. Our work then provides the extension of the Fourier-Mukai transform constructed by Arinkin and Melo-Rapagnetta-Viviani, which corresponds to autoduality of $\mathrm{GL}(n,\mathbb{C})$-Hitchin fibres in this class of singular spectral curves.

Theorems & Definitions (41)

• Theorem 2.1: Theorem A melo0
• Theorem 2.2
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4
• proof