Moduli spaces of sheaves and symplectic varieties

• Applicant: Dr. Markus Zowislok
• Subject Area: Mathematics
• Term: from 2012 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 223873773

The first goal of the research programme is a better understanding of the geometry of moduli spaces of semistable sheaves on projective K3 or abelian surfaces as well as the connection to the geometry of moduli spaces of semistable sheaves on projective Enriques and bielliptic surfaces, respectively. All these surfaces are projective complex surfaces of Kodaira dimension 0. They are real 4-dimensional manifolds, and they show a very rich geometry. Sheaves on these surfaces are e.g. vector bundles, and the mentioned moduli space is a classifying space containing representatives of a significant subclass of sheaves. In the case of K3 and abelian surfaces the moduli spaces are symplectic if the ample divisor is general. By resolving symplectically if necessary, and by taking a certain fibre in the abelian surface case, one gets examples of projective hyperkähler manifolds. All known examples of such manifolds are deformation equivalent to these examples. I am particularly interested in nongeneral ample divisors. If one considers Enriques surfaces, then the universal covering is a K3 surface. This relation transfers to an interesting relation between the corresponding moduli spaces. There is an analogous relation for abelian and bielliptic surfaces. One important special case of a moduli space of sheaves is the Hilbert scheme of points. Such a Hilbert scheme resolves singularities of symmetric products of surfaces. If one considers certain special sheaves that are equivariant with respect to a reductive group, so-called constellations, then their moduli space is a good candidate for a resolution of singularities of the geometric invariant theoretic group quotient. The second goal is the investigation of its properties and potential.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Host: Professor Dr. Richard Thomas