Simple vector bundles on plane degenerations of an elliptic curve

Abstract

Abstract. In 1957 Atiyah classified simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classification by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that a simple vector bundle on such a curve is completely determined by its rank, multidegree and determinant. Our approach, based on the representation theory of boxes, also yields an explicit description of the corresponding universal families of simple vector bundles.

Vector bundles on degenerations of elliptic curves and Yang–Baxter equations

“…Taking their composition, we obtain a meromorphic section (r 13 13 ) 13 (r 32 12 ) 12 of the holomorphic vector bundleĥ * 1 Hom(π * 2 P,π * 1 P)⊗ĥ * 2 Hom(π * 3 P,π * 2 P)⊗ĥ * 3 Hom(π * 1 P,π * 3 P). In a similar way, two other meromorphic sections (r 12 12 ) 12 (r 13 23 ) 23 and (r 23 23 ) 23 (r 12 13 ) 13 of this vector bundle can be defined. These sections are holomorphic overB.…”

“…• Ifr 0 (y) does not have infinitesimal symmetries and if s(v; y) is another solution of (12) of the form (10) and such thatr 0 (y) = pr ⊗ pr (s 0 (y)), then there exist α 1 ∈ C * and α 2 ∈ C such that s(v; y) = α 1 exp(α 2 vy)r(v; y [27]. Moreover, it was proven by Polishchuk in [56] that any elliptic solution of the classical Yang-Baxter equation (3) can be lifted to a solution of (12) having a Laurent expansion of the form (10). However, Schedler showed in [59] that there exist trigonometric solutions of (CYBE), which can not be lifted to a solution of the associative Yang-Baxter equation (12) of the form (10).…”

“…The remaining cases, i.e. the Kodaira fibers of type I 2 , I 3 , III and IV, were considered in a recent paper of Bodnarchuk, Drozd and Greuel [12]. Conjecture 6.…”

“…Let g be a simple complex Lie algebra (throughout this paper g = sl n (C)), , : g × g → C the Killing form. The classical Yang-Baxter equation is (1) [r 12 (y 1 , y 2 ), r 23 (y 2 , y 3 )] + [r 12 (y 1 , y 2 ), r 13 (y 1 , y 3 )] + [r 13 (y 1 , y 3 ), r 23 (y 2 , y 3 )] = 0, where r(x, y) is the germ of a meromorphic function of two complex variables in a neighbourhood of 0, taking values in g ⊗ g. A solution of (1) is called unitary if r 12 (y 1 , y 2 ) = −r 21 (y 2 , y 1 ) and non-degenerate if r(y 1 , y 2 ) ∈ g ⊗ g ∼ = g * ⊗ g ∼ = End(g) is invertible for generic (y 1 , y 2 ). On the set of solutions of (1) there exists a natural action of the group of holomorphic function germs φ : (C, 0) −→ Aut(g) given by the rule (2) r(y 1 , y 2 ) → φ(y 1 ) ⊗ φ(y 2 ) r(y 1 , y 2 ).…”

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

“…The case of a nodal Weierstraß curve has been treated by the first-named author in [9], the corresponding result for a cuspidal cubic curve is due to Bodnarchuk and Drozd [7]. The remaining cases (Kodaira fibers of type I 2 , I 3 , III and IV) are due to Bodnarchuk, Drozd and Greuel [8]. Their method actually allows to prove this theorem for arbitrary Kodaira cycles of projective lines.…”