On the semistability of certain Lazarsfeld–Mukai bundles on abelian surfaces

Let X= J(C~) , the Jacobian of a genus 2 curve C~ over C, and let Y be the associated Kummer surface. Consider an ample line bundle L= O(mC~) on X for an even number m, and its descent to Y, say L′. We show that any dominating component of Wd1(|L′|) corresponds to μL′-stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve C∈ | L| and a base-point free gd1 on C, say (A, V), we study the μL-semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (A, V)) on X. Under certain assumptions on C and the gd1, we show that the above Lazarsfeld–Mukai bundles are μL-semistable.