×

Vector bundles and the Icosahedron. (English) Zbl 1227.14036

GarcĂ­a-Prada, Oscar (ed.) et al., Vector bundles and complex geometry. Conference on vector bundles in honor of S. Ramanan on the occasion of his 70th birthday, Madrid, Spain, June 16–20, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4750-3/pbk). Contemporary Mathematics 522, 71-87 (2010).
The classification of vector bundles over smooth elliptic curves was performed long ago by Atiyah, who described the three possible cases and their relations. If \(f\) is a harmonic cubic form in three variables, (and thus \(f\) is a solution to Laplace’s equation), one has a natural Pfaffian representation of \(f\), which in turn determines a rank two bundle \(E\), over the curve \(C\) defined by \(f\). The author shows how the class of \(E\) in Atiyah’s classification is determined by the geometry of the curve \(C\). More precisely, a harmonic cubic contains two icosahedral subsets of points, forming the vertexes of a regular icosahedron. Sextuplets of points of this type are parametrized by the Mukai-Unemura threefold \(Z\), which parametrizes isotropic subspaces of dimension \(3\) inside the \(7\)-dimensional subspace of harmonic cubic polynomials. The author describes points of \(Z\) corresponding to icosahedral sets belonging to harmonic curves, whose rank \(2\) bundle \(E\) fits in specific classes of Atiyah’s classification.
For the entire collection see [Zbl 1197.14001].

MSC:

14H52 Elliptic curves
14H60 Vector bundles on curves and their moduli
14M12 Determinantal varieties

Keywords:

elliptic curves
PDFBibTeX XMLCite
Full Text: arXiv