Holomorphic ${\rm sl}(2, \mathbb{C})$-differential systems on Riemann surfaces and curves in compact quotients of ${\rm SL}(2,\mathbb{C})$
Sorin Dumitrescu
Université Côte d'Azur, Nice, France
Abstract:
We explain the strategy of a
recent result that constructs holomorphic ${\rm sl}(2, \mathbb{C})$--differential systems over
some Riemann surfaces $\Sigma_g$ of genus $g \,\geq\, 2$, such that the image of the associated
monodromy homomorphism is some cocompact Kleinian subgroup
$\Gamma \, \subset \, {\rm SL}(2, \mathbb{C})$.
As a consequence,
there exist holomorphic maps from $\Sigma_g$ to the quotient ${\rm SL}(2, \mathbb{C})/
\Gamma$, that do not factor through any elliptic curve. This answers positively
a question asked by Huckleberry and Winkelmann, also raised by Ghys.
This is a joint work with Indranil Biswas (TIFR, Mumbai), Lynn Heller (BIMSA, Beijing) and Sebastian Heller (BIMSA, Beijing).