EPW varieties as moduli spaces on ordinary GM surfaces and special GM threefolds

Abstract: We show that the double dual EPW sextic and double EPW sextic associated with a strongly smooth Gushel--Mukai surface can be realized as moduli spaces of semistable objects with respect to a stability condition on the bounded derived category of it. Also, we observe that double dual EPW surface and double EPW surface associated with a special Gushel--Mukai threefold can be realized as moduli spaces of semistable objects on its Kuznetsov component. As an application, we refine a statement of Bayer and Perry about Gushel--Mukai threefolds with equivalent Kuznetsov components for the special ones.