2013/10/29 by Marco Spinaci, Spinaci, Marco
Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.CV #math.DG
paper · pdf · doi:10.48550/arxiv.1310.7694
32 pages. Several typos have been corrected and some references have been added. To appear on Math. Z
arxiv created 2014/05/09 · arxiv updated 2014/05/12
We study the deformations of twisted harmonic maps f with respect to the representation ρ. After constructing a continuous "universal" twisted harmonic map, we give a construction of every first order deformation of f in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical points of the energy functional E coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is a potential for the Kähler form of the "Betti" moduli space; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of E at critical points.