Nilpotent operators and weighted projective lines

Abstract

Abstract. We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those.The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2, 3, p), obtained by Serre construction from the triangle singularity x 2 + y 3 + z p . For p = 6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2, 3, 6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve.Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.

Dimension Vectors of Indecomposable Objects for Nilpotent Operators of Degree 6 with One Invariant Subspace

“…For each i = 1, 2, 3, the set {S i,j } j ∈Z p i forms a full list of consecutive with respect to τ −1 X quasi-simple objects in the exceptional tubes in C ∞ . The families {S i,j } yield another ordered basis (a) the sequences rk(T ) = (rk(T i )) i∈ [10] , deg(T ) = (deg(T i )) i∈ [10] and μ(T ) = (μ(T i )) i∈ [10] of ranks, degrees and slopes of indecomposable direct summands…”

“…One can show that if T is a tilting bundle, then the equality [rk(T l )] l∈ [10] = h 1 (:= [10] = h ∞ . In fact, these two equalities follow from some formula which holds for tilting sheaves over all weighted projective lines of tubular type, and can be treated as a tool for finding rk(T ) and deg(T ) of potential "tilted realization" of a given tubular algebra in general situations (this will be discussed in a subsequent publication).…”

“…We start by a rather general fact which says how to determine, in some nice situations, the Cartan matrix of the algebra End X ( i∈ [10] T i ) via the matrix of the respective values of the Euler form. …”

“…Using completely different methods in [10] it was shown that the stable category of S(ñ) is equivalent to the stable category of vector bundles over a weighted projective line of weight type (2, 3, n) in the sense of [7]. Here the notion of a stable category of vector bundles means that in the category of vector bundles we factor out the ideal of all morphisms which factor through finite direct sums of line bundles.…”

“…Here the notion of a stable category of vector bundles means that in the category of vector bundles we factor out the ideal of all morphisms which factor through finite direct sums of line bundles. The investigation of such categories and their connections to categories of nilpotent operators with chains of invariant submodules was continued in [11] and [12].…”