Nilpotent operators and weighted projective lines

TL;DR

This paper reveals a deep connection between singularity theory, vector bundles on weighted projective lines, and the invariant subspace problem of nilpotent operators, providing new insights and tools for longstanding mathematical challenges.

Contribution

It introduces a novel approach linking nilpotent operators to vector bundles on weighted projective lines, extending classification results and establishing triangulated categories with Calabi-Yau properties.

Findings

Main results of Ringel and Schmidmeier derived from vector bundle properties

Classification of vector bundles for type (2,3,6) covers the p=6 case

Construction of a sequence of triangulated Calabi-Yau categories

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.