Schottky uniformization and vector bundles over Riemann surfaces

We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient condition for this map to be a submersion, when restricted to representations producing stable bundles. Using a generalized version of Riemann's bilinear relations, this condition is shown to be true on the subspace of unitary Schottky representations.Comment: 16 pages; AMSLate

Higgs bundles and representation spaces associated to morphisms

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\subset G$ be a maximal compact subgroup. Let $X , Y$ be irreducible smooth complex projective varieties and $f: X \rightarrow Y$ an algebraic morphism, such that $\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_* : \pi_1(X) \rightarrow\pi_1(Y)$ is surjective. Define $${\mathcal R }^f(\pi_1(X),\, G)\,=\, \{\rho\, \in\, \text{Hom}(\pi_1(X),\, G)\, \mid\, A\circ\rho \ \text{ factors through }~ f_*\}\, ,$$ $${\mathcal R }^f(\pi_1(X),\, K)\,=\, \{\rho\, \in\, \text{Hom}(\pi_1(X),\, K)\, \mid\, A\circ\rho \ \text{ factors through }~ f_*\}\, ,$$ where $A: G \rightarrow \text{GL}(\text{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(\pi_1(X, x_0), G)/\!\!/G$ admits a deformation retraction to ${\mathcal R }^f(\pi_1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$

Geometria e Topologia

Editor Convidado: Carlos Florentino Rui Albuquerque, "Geometria dos espaços de gwistor" Ethan Cotterill, "Counting maps from curves to projective space via graph theory" Rosa Sena-Dias, "Spectral theory for toric orbifolds"Pedro Macias Marques, Helena Soares, "Monads on Segre varieties" Leonor Godinho, Silvia Sabatini, "Towards classifying Hamiltonian torus actions with isolated fixed points" Pedro Vaz, "On Jaeger’s HOMFLY-PT expansions, branching rules and link homology: a progress report" Orlando Neto, Pedro C. Silva, "Waldhausen decomposition and Systems of PDEs"