16 research outputs found
On totally geodesic submanifolds in the Jacobian locus
We study submanifolds of A_g that are totally geodesic for the locally
symmetric metric and which are contained in the closure of the Jacobian locus
but not in its boundary. In the first section we recall a formula for the
second fundamental form of the period map due to Pirola, Tortora and the first
author. We show that this result can be stated quite neatly using a line bundle
over the product of the curve with itself. We give an upper bound for the
dimension of a germ of a totally geodesic submanifold passing through [C] in
M_g in terms of the gonality of C. This yields an upper bound for the dimension
of a germ of a totally geodesic submanifold contained in the Jacobian locus,
which only depends on the genus. We also study the submanifolds of A_g obtained
from cyclic covers of the projective line. These have been studied by various
authors. Moonen determined which of them are Shimura varieties using deep
results in positive characteristic. Using our methods we show that many of the
submanifolds which are not Shimura varieties are not even totally geodesic.Comment: To appear on International Journal of Mathematic
The mapping class group and the Meyer function for plane curves
For each d>=2, the mapping class group for plane curves of degree d will be
defined and it is proved that there exists uniquely the Meyer function on this
group. In the case of d=4, using our Meyer function, we can define the local
signature for 4-dimensional fiber spaces whose general fibers are
non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our
local signature will be given.Comment: 24 pages, typo adde
Complements of hypersurfaces, variation maps and minimal models of arrangements
We prove the minimality of the CW-complex structure for complements of
hyperplane arrangements in by using the theory of Lefschetz
pencils and results on the variation maps within a pencil of hyperplanes. This
also provides a method to compute the Betti numbers of complements of
arrangements via global polar invariants