Stability of syzygy bundles of Ulrich bundles

TL;DR

This paper proves that the syzygy bundle associated with an initialized Ulrich bundle on certain smooth varieties is semistable, extending understanding of bundle stability in algebraic geometry.

Contribution

It establishes the semistability of syzygy bundles of Ulrich bundles on specific classes of smooth varieties, a new result in the study of vector bundle stability.

Findings

Syzygy bundle S(E) is semistable on smooth K3 surfaces and Fano varieties under given conditions.

Ulrich bundles' syzygy bundles exhibit stability properties in the specified geometric contexts.

The result applies to varieties with index greater than n-3, broadening stability results in algebraic geometry.

Abstract

Let X be either a smooth K3 surface or a smooth Fano variety (i.e. $-K_X$ is ample) of dimension $n$ and index $i_X> n-3$ and let E be an initialized Ulrich bundle on X. In this paper, we show that the syzygy bundle $S(E)$, defined as the kernel of the evaluation map $H^0(X,E)\otimes O_{X}\rightarrow E$, is semistable.