  We describe a quasi-linear algorithm for computing Igusa class polynomials of
Jacobians of genus 2 curves via complex floating-point approximations of their
roots. After providing an explicit treatment of the computations in quartic CM
fields and their Galois closures, we pursue an approach due to Dupont for
evaluating $\theta$- constants in quasi-linear time using Newton iterations on
the Borchardt mean. We report on experiments with our implementation and
present an example with class number 17608.
