arXiv subjects
Publications and source records attributed to Peter Dalakov.
We obtain explicit formulae for the Donagi-Markman (Bryant-Griffiths, Yukawa) cubic for Hitchin systems of type $A_2$, $B_2$ and $G_2$. This is achieved by evaluating the quadratic residues in the Balduzzi-Pantev formula, using a previous result of ours. For $G_2$ we also recover earlier results of Hitchin.
In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss--Manin) derivative of the Seiberg--Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration. Dedicated to Tony Pantev on the occasion of his 60th birthday.
These are largely expanded notes from lectures on Higgs moduli and abelianisation given in Angers, France (2014) and Guaruja, Brazil (2015). Dedicated to Ugo Bruzzo on his 60-th birthday. Version 2: minor corrections.
The present note is mostly a survey on the generalised Hitchin integrable system and moduli spaces of meromorphic Higgs bundles. We also fill minor gaps in the existing literature, outline a calculation of the infinitesimal period map and review briefly some related geometries.
Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We prove that the Balduzzi--Pantev formula holds along maximal rank symplectic leaves of the G-generalised Hitchin system.
Recently, E.Martinengo obtained results on obstructions to deformations of Higgs pairs by describing an L-infinity morphism inducing the Hitchin map. In this note we show that analogous results hold for principal G-Higgs bundles, where G is a complex reductive group. We show that the L-infinity morphism has a Lie-algebraic analogue inducing the adjoint quotient morphism.
Fix a simple complex Lie group G and a principal sl(2,C) subalgebra of Lie(G). Then the moduli space of semi-stable, topologically trivial G-Higgs bundles on a hyperbolic, spin Riemann surface acquires a marked point. This is the unique C*-fixed point on the Hitchin section. We describe a universal analytic family of deformations which provides holomorphic Darboux coordinates in a neighbourhood of the section. This is a special case of a more general deformation-theoretic construction in the spirit of Kuranishi theory. As a toy example of the latter we consider the tautological family of centralisers over the Kostant slice.