Orlov spectra: bounds and gaps

• 成果类型: Article
• 署名作者: Ballard, Matthew; Favero, David; Katzarkov, Ludmil
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; University of Vienna
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0367-y
• 发表日期: 2012
• 页码: 359-430
• 关键词: triangulated categories; generating hypothesis; representation dimension; coherent sheaves; split-sequences; SINGULARITIES; MODULES; ghosts; ring

摘要

The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov, building on work of A. Bondal-M. Van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a generator, by closing the object under a certain monodromy action, and uniformly bound this generator's generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one.