Categorical base loci and Orlov spectra

In this proposal we initiate the theory of categorical base loci. We introduce the idea of building the following analogies: 1) The analogy between classical linear system and certain functor of a category. 2) The second analogy is between base loci of a linear system and categorical base loci. We propose a theory of categorical multiplier ideal sheaves. Our approach is based on the pioneering works by Seidel Ein, Lazarsfeld, Mustata, Nakamaye, Popa, Budur The main ideas of the project is that the gaps in the Orlov spectra can be seen as jump numbers of a multiplier ideal sheaf. If completed this idea will lead to the solution of some classical questions in Algebraic Geometry.

The study of Geometry in the 20th Century was devoted, in large part and with astounding success, to the classification and parametrization of geometrical objects. However, these objects, of various kinds, were uniformly viewed somehow as sets of points. Along the way, the relationship with categorical structures grew steadily, With Kontsevich`s introduction of HMS - Homological Mirror Symmetry, a subtle change was introduced, in that Geometry began to be seen within a categorical structure. And the concurrent development of the theory of higher stacks meant that geometric structures were no longer viewed just as sets of points but rather as objects enclosing a higher structure. Within this project are introduced and developed new structures and computed many examples: (1) Categorical Kaehler Geometry (2) Categories with a phase gap and norms on them (3) Topology on the class of categories with a phase gap (4) Large class of new categorial invariants: non-commutative curve-counting invariants are introduced. These novelties open new perspectives not only in non-commutative geometry but also new connections to number theory, combinatorics, classical geometry.