  In this paper we prove that finite index subgroups of genus 3 mapping class
and Torelli groups that contain the group generated by Dehn twists on bounding
simple closed curves are not Kahler. These results are deduced from explicit
presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli
groups and of the relative completions of genus 3 mapping class groups. The
main results follow from the fact that these presentations are not quadratic.
To complete the picture, we compute presentations of completed Torelli and
mapping class in genera > 3; they are quadratic. We also show that groups
commensurable with hyperelliptic mapping class groups and mapping class groups
in genera < 3 are not Kahler.

Algebraic Geometry;Complex Variables;Geometric Topology