Hebrew University topology and geometry seminar

Abstract: Consider the collection of line bundles O,O(1),...,O(n) on n-dimensional complex projective space and let A be the endomorphism ring of its direct sum. A seminal result of Beilinson states that the bounded derived category of coherent sheaves on P^n is equivalent to the bounded derived category of right modules over A. A fundamental question asks which toric manifolds admit "exceptional" collections of line bundles, generalizing Beilinson's example? Recently, examples of toric manifolds which do not admit (full strongly) exceptional collections of line bundles were found by Hille & Perling and by Efimov, disproving previous conjectures on the subject.

In this talk we would take a "mirror approach" to the question. We consider toric Fano manifolds. For such manifolds the theory of quantum cohomology associates a system of algebraic equations, known as the Landau-Ginzburg system of X (first introduced by V. Batyrev). The aim of the talk is to show that in low dimensions (two and three) there are some striking relations between the solution set of the LG-system (denoted Crit(X)) and known examples of exceptional collections in Pic(X). For the considered examples we shall (a) introduce a map L: Crit(X) ---->Pic(X) whose image is an exceptional collection (b) show that, under the map L, monodromies of the Landau-Ginzburg system turn to related to quiver representations of the corresponding collection. These relations lead to postulate on the general, higher dimensional, case.