Derived categories of symmetric products and moduli spaces of vector bundles on a curve

TL;DR

This paper demonstrates embeddings of derived categories from symmetric products of curves into those of moduli spaces of vector bundles, supporting conjectures about their semiorthogonal decompositions and revealing new Fano visitor properties.

Contribution

It establishes the embedding of derived categories of symmetric products into moduli spaces, providing evidence for semiorthogonal decompositions and identifying new Fano visitors among abelian varieties.

Findings

Derived categories of symmetric products embed into moduli space categories.

Supports semiorthogonal decomposition predictions for moduli spaces.

Identifies Jacobians and certain abelian varieties as Fano visitors.

Abstract

We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal decomposition of the derived category of the moduli space, expected by a motivic computation. As an application, we show that all Jacobian varieties, symmetric products of curves and all principally polarized abelian varieties of dimension at most three, are Fano visitors. We also obtain similar results for motives.