89 research outputs found
Uniformly effective Shafarevich Conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus
The paper's main result is an effective uniform bound for the finiteness
statement of the Shafarevich Conjecture over function fields. Several results
on the projective geometry of curves are established in the course of the
proof. These results should be of independent interest. As a corollary, a
uniform effective bound for the Mordell Conjecture over function fields is
derived via Parshin's trick.Comment: 19 page
On projectivized vector bundles and positive holomorphic sectional curvature
We generalize a construction of Hitchin to prove that, given any compact
K\"ahler manifold with positive holomorphic sectional curvature and any
holomorphic vector bundle over , the projectivized vector bundle
admits a K\"ahler metric with positive holomorphic sectional
curvature
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