Project A8: Quasi-hereditary algebras, toric geometry and derived categories



Principal investigators

Participating scientists

Summary

Derived categories relate categories of modules with categories of coherent sheaves via tilting. It was recently shown that any rational surface admits a tilting object. Moreover, under such a tilting equivalence several invariants are preserved. It turned out only recently, that quasi-hereditary algebras, a class of algebras coming from highest weight categories, play an important role under those equivalences. Moreover, toric geometry became important for tilting via toric systems. The aim of this project is to study invariants for derived categories related to algebra and the underlying geometry. A first project studies algebraic surfaces, in particular spherical twists and perfect tilting. Moreover, we want to generalize the toric systems, defined only for surfaces so far, to higher dimensions. Some particular quasi-hereditary algebras, playing a role for parabolic group actions, occur as endomorphism algebras of partial tilting bundles. This allows to use the classification of tilting modules for classifying certain partial tilting bundles on the corresponding surface.

Moduli spaces of quiver representations also play an important role and relate to well-known moduli spaces in algebraic geometry. So far, only the toric ones are well understood. We also intend to study the derived category of these moduli spaces. Besides studying these derived categories as interesting invariants by themselfes, one can also look at their Grothendieck groups and distinguished elements therein (like e.g. tilting objects) as the invariants for their study in algebra and geometry. Another source of such distinguished elements in the derived category or the Grothendieck group of coherent sheaves in the complex algebraic geometric context come from the theory of characteristic classes of singular spaces in the form of graded objects associated to suitable filtered De Rham complexes. In the presence of a compatible duality functor one can also study the corresponding Witt groups, e.g. for the derived category of constructible or perverse sheaf complexes. with the corresponding (twisted) intersection cohomology complexes as distinguished selfdual objects. Similarly for highest weight categories with a duality, with the corresponding simple objects as distinguished selfdual objects. A final source of examples in the context of equivariant derived categories of coherent sheaves comes from the theory of staggered sheaves, which for toric varieties often yields a selfdual t-structure.
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