HRR Polynomials and ASF Systems

In the first part of this thesis we calculate the Huybrechts-Riemann-Roch polynomial for six-dimensional hyperkahler manifolds following Debarre. These are the Riemann-Roch polynomials written in terms of the Beauville-Bogomolov form. Many invariants are encoded into these polynomials, such as the Euler characteristic and the Fujiki constant. Whilst some has been calculated in Debarre's paper, our method uses Rozansky-Witten invariants along with Fujiki relations, and it works for any values of q(L,F). Assuming a conjecture from Sawon, the first two possible polynomials are Hilb^3 and Kum_3, two of the known deformation types of hyperkahler manifolds. We then use method from Debarre to calculate the cases when a=3,4. In the second part of this thesis we look at the Arbarello-Saccà-Ferretti systems, which are relative Prym varieties. In particular they are a collection of Lagrangian fibrations on holomorphic symplectic varieties that were constructed by Arbarello, Saccà and Ferretti. We calculate some invariants of the ASF system in the lowest dimensional non-hyperelliptic case, when g=3 and the ASF system has dimension four. We calculated the Euler characteristic, the number of singularities (which are known to be of type C^4/+-1 from the work of Arbarello, Saccà and Ferretti) and the degree of the discriminant locus parametrising singular fibres. These calculations suggest a connection to a Lagrangian fibration on another symplectic orbifold, which arise as a quotient of Hilb^2S by a symplectic action of the Klein four group. We conjecture that this is a specialisation of the ASF system.