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Zbl 0758.47038
Previato, Emma; Wilson, George
Differential operators and rank 2 bundles over elliptic curves. (English)
[J] Compos. Math. 81, No.1, 107-119 (1992). ISSN 0010-437X; ISSN 1570-5846/e

Let $(L,M)$ be a commuting pair of linear ordinary differential operators, whose coefficients are analytic functions of a variable $x$. The set of pairs $(a,b)$ which are eigenvalues for the same eigenfunction $\Phi$ of $L$, $M$, defines an affine curve $X\sb 0\subset\bbfC\sp 2$; when $X\sb 0$ is smooth, the spaces of eigenfunctions for all $(a,b)\in X\sb 0$ glue together giving a vector bundle $E\sb 0$; $X\sb 0$ can be completed to a projective curve $X$, adding one point; when $X$ is smooth, also the dual of $E\sb 0$ can be completed to a vector bundle $E$ on $X$.\par The authors are concerned here with the classifications of the smooth elliptic curves $X$ and rank 2 vector bundles $E$ arising from the previous construction. Such a classification was yet given before, but not when the starting point is a commuting pair $(L,M)$ which, under the action of the automorphism group of the algebra of differential operators is normalized in a `standard' way (i.e. $L$ has leading coefficient 1 and second coefficient 0) which is suitable for the applications to partial differential equations. In this setting, only the elliptic curve that one gets from the previous construction were classified by Grünbaum. The authors describe here the rank 2 bundles obtained from a normalized commuting pair $(L,M)$ over a smooth elliptic curve $X$; they show that one obtains indecomposable and decomposable bundles, even of the type ${\cal O}\sb P\sp 2$, $(P\in X)$, which was not detected in some previous classification.
[L.Chiantini (Napoli)]

MSC 2000:: *47E05 Ordinary differential operators
14H60 Vector bundles on curves
14H52 Elliptic curves
14F10 Special sheaves

Keywords: commuting pair of linear ordinary differential operators; vector bundle; classifications of the smooth elliptic curves; automorphism group; algebra of differential operators; indecomposable and decomposable bundles

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