Any commutative subalgebra A in the algebra of ordinary differential operators admits a natural geometric invariant consisting of an irreducible (possibly singular) projective curve C (called spectral curve) and a semi-stable torsion free sheaf ℱ on it (called spectral sheaf). In the case the rank of A is one (meaning that A contains a pair of differential operators of mutually prime orders), the algebra A can be recovered from its spectral datum (C, ℱ) (Krichever correspondence).
All commutative subalgebras of ordinary differential operators of genus one and rank two were classified in the 80ies by Krichever, Novikov and Gruenbaum. It is a natural problem to describe the spectral sheaves of such algebras. This problem was solved by Previato and Wilson in the case the spectral curve is smooth, their answer was given in terms of Atiyah’s classification of vector bundles on an elliptic curve. However, the case of a singular spectral curve remained opened.
In my talk (based on a joint work with Alexander Zheglov: arXiv:1602.08694) I shall explain how the Fourier-Mukai transform allows to describe the spectral sheaf of a genus one commutative subalgebra of ordinary differential operators. As a by-product, I shall also show how the low rank objects of the category of semi-stable sheaves on a cuspidal Weierstrass cubic curve (known to be representation wild) can be classified.
This video was produced by Syracuse University Department of Mathematics as part of ICRA 2016.
