Admissible subcategories in derived categories of moduli of vector bundles on curves

TL;DR

This paper demonstrates that the derived category of a high-genus curve can be fully faithfully embedded into the derived category of its moduli space of vector bundles, and establishes a semi-orthogonal decomposition involving this embedding.

Contribution

It generalizes previous results by embedding the derived category of the curve into that of the moduli space for higher rank bundles and constructs a semi-orthogonal decomposition.

Findings

Fully faithful embedding of the derived category of the curve into the moduli space's derived category.

Construction of a semi-orthogonal decomposition involving the embedded subcategory.

Extension of known results from rank 2 to higher ranks.

Abstract

We show that the Poincar\'e bundle gives a fully faithful embedding from the derived category of a curve of sufficiently high genus into the derived category of its moduli space of bundles of rank $r$ with fixed determinant of degree 1. Moreover we show that a twist of the embedding, together with 2 exceptional line bundles, gives the start of a semi-orthogonal decomposition. This generalises results of Narasimhan and Fonarev-Kuznetsov, who embedded the derived category of a single copy of the curve, for rank 2.