Complex Moduli Space Research Articles

Pseudo-real principalG-bundles over a real curve

We consider stable and semistable principal bundles over a smooth projective real algebraic curve, equipped with a real or pseudo-real structure in the sense of Atiyah. After fixing suitable topological invariants, one can build a suitable gauge theory, and show that the resulting moduli spaces of pseudo-real bundles are connected. This in turn allows one to describe the various fixed point varieties on the complex moduli spaces under the action of the real involutions on the curve and the structure group.

On the symplectic eightfold associated to a Pfaffian cubic fourfold

Abstract We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z → Hilb 4 ⁡ ( X ) Z\to\operatorname{Hilb}^{4}(X) , where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points. This note is an appendix to http://dx.doi.org/10.1515/crelle-2014-0144.

Topology of moduli spaces of free group representations in real reductive groups

Abstract Let G be a real reductive algebraic group with maximal compact subgroup K, and let Fr be a rank r free group. We show that the space of closed orbits in Hom(Fr ,G)/G admits a strong deformation retraction to the orbit space Hom(Fr ,K)/K. In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups G, such as Sp(4,ℝ) and U(2,2). We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

Second variation of Zhang’s λ-invariant on the moduli space of curves

We compute the second variation of the $\lambda$-invariant, recently introduced by S. Zhang, on the complex moduli space ${\cal M}_g$ of curves of genus $g \geq 2$, using work of N. Kawazumi. As a result we prove that $(8g+4)\lambda$ is equal, up to a constant, to the $\beta$-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the $\lambda$-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form.