Coherent sheaves on abelian varieties: moduli, derived categories and stability conditions

Theoretical physics has come up with a speculative model for the universe, in which four-dimensional space-time is enriched by additional "puny" dimensions, forming a complex three dimensional manifold of size the order of the Planck length. The project is a purely mathematical study of questions arising when this manifold forms a complex torus. A main theme carries the name Donaldson-Thomas invariants: a loose analogy to the "number of elementary particles" in physics. Mathematically, they represent counts of certain classes of geometric objects such as curves or vector bundles. Over simplified calculations suggest there are finitely many such objects, making it meaningful to count them. In reality however they are far from finite, rather they form new spaces of high dimension. Donaldson-Thomas invariants are a precise replacement for the "count" in this infinite landscape, which has been little studied previously in the complex torus situation.

- Det primære er konkrete forskningsresultater innenfor prosjektets tematikk. Dette er oppnådd i omtrent samme grad som forventet, i form av publikasjoner i fagfellevurderte tidsskrifter samt foredrag ved seminarer og workshops. - Prosjektet finansierer én doktorgradskandidat. Vedkommende har i skrivende stund ikke disputert, men er i avslutningsfasen med mål om å levere inneværende semester. - Prosjektet har i stor grad bidratt til å utvikle et algebraisk-geometri-miljø ved UiS av høy kvalitet og med internasjonal anerkjennelse. To workshops i Stavanger finansiert av prosjektet har vært av vesentlig betydning for det siste.

Moduli spaces for stable sheaves and complexes on higher dimensional varieties, and indeed the very concept of stability, plays an increasingly central role in algebraic geometry. One driving force, and from a mathematical point of view a valuable resourc e of fruitful ideas, is string theory and mirror symmetry in mathematical physics. In this regard, the case of varieties with trivial canonical bundle is central. Purely mathematical motivation comes from treating moduli spaces for sheaves as a source of nontrivial, and to some extent accessible, examples in higher dimensional geometry. Also from this point of view, varieties with trivial canonical bundle give rise to particularly interesting geometries. Coherent sheaves on higher dimensional projective manifolds are poorely understood, and this is true also for abelian threefolds. The fundamental idea in this project is to study the questions alluded to above in the case of abelian varieties, and in particular abelian threefolds. Central questions are t o describe the geometry in explicit examples of moduli spaces for sheaves and complexes on abelian varieties, to adapt the theory of generalized and categorified Donaldson-Thomas invariants to abelian threefolds, to describe Bridgeland's stability manifol d in this case, to study wall crossing phenomena and the associated behaviour of Donaldson-Thomas invariants and the birational maps between moduli spaces. In the situation where the underlying threefold is Calabi-Yau with vanishing irregularity, there i s a very active research community addressing these topics. For the questions regarding Donaldson-Thomas invariants, taking abelian threefolds instead is somewhat unorthodox. We believe, and want to demonstrate, that a concentrated study of the abelian si tuation will be fruitful. This seems to be a timely problem, as the first example of a Bridgeland stability condition in the abelian threefold case has just been established.