Several important geometric structures can be constructed and studied via their *twistor space*, i.e. as a parameter space of (real) rational curves in a complex manifold. Naturally arising examples of these geometries, which include hyperkähler metrics, are of great significance in several branches of mathematics and mathematical physics: e.g. quiver varieties in representation theory, Hitchin's moduli spaces in algebraic geometry and integrable systems theory, gauge-theoretic moduli spaces of monopoles and instantons in mathematical physics.

Many of the above-mentioned examples can be constructed as moduli spaces of higher genus curves in a complex 3-fold equipped with an antiholomorphic involution. The aim of this project is to investigate the global geometry of Kodaira moduli spaces via twistor methods; more precisely, the project aims to investigate completeness and the asymptotic geometry of natural metrics on manifolds arising as smooth loci of Hilbert schemes of real algebraic curves (satisfying certain stability conditions) in complex (non-compact) manifolds, particularly in 3-folds.

## Publications

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the canonical Obata connection on the Hilbert scheme of cohomologically stable twisted rational curves of degree *d* in the ℙ3∖ℙ1 is flat for any *d*≥3.

**Related project(s):****7**Asymptotic geometry of moduli spaces of curves

We investigate the geometry of the Kodaira moduli space* M* of sections of a twistor projection, the normal bundle of which is allowed to jump. In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of *M* extends to a logarithmic connection on *M*.

**Related project(s):****7**Asymptotic geometry of moduli spaces of curves

We describe the natural geometry of Hilbert schemes of curves in projective spaces.

**Related project(s):****7**Asymptotic geometry of moduli spaces of curves

## Team Members

**Prof. Dr. Roger Bielawski**

Project leader

Leibniz-Universität Hannover

bielawski(at)math.uni-hannover.de

** Carolin Peternell**

Researcher

Leibniz-Universität Hannover

peternell(at)math.uni-hannover.de