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Abstract
Let [Formula: see text] be a compact connected Riemann surface of genus at least two. The Abel–Jacobi map [Formula: see text] is an embedding if [Formula: see text] is less than the gonality of [Formula: see text]. We investigate the curvature of the pull-back, by [Formula: see text], of the flat metric on [Formula: see text]. In particular, we show that when [Formula: see text], the curvature is strictly negative everywhere if [Formula: see text] is not hyperelliptic, and when [Formula: see text] is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of [Formula: see text] fixed by the hyperelliptic involution.
Highlights
Let X be a compact connected Riemann surface of genus g, with g ≥ 2
The gonality of X is defined to be the smallest integer γX such that there is a nonconstant holomorphic map from X to CP1 of degree γX
Picd(X) is equipped with a flat Kahler form, which we will denote by ω0
Summary
Let X be a compact connected Riemann surface of genus g, with g ≥ 2. Picd(X) is equipped with a flat Kahler form, which we will denote by ω0. Φ∗ω0 is a Kahler form on Symd(X), whenever d < γX. Our aim here is to study the curvature of this Kahler form φ∗ω0 on Symd(X). Consider the g–dimensional vector space H0(X, KX ) consisting of holomorphic oneforms on X. It is equipped with a natural Hermitian structure. We prove that the holomorphic Hermitian vector bundle ρ∗V −→ Symd(X) is isomorphic to the holomorphic cotangent bundle ΩSymd(X) equipped with the Hermitian structure given by φ∗ω0 (Theorem 3.1). Since the curvature of the holomorphic Hermitian vector bundle V −→ G is standard, Theorem 3.1 gives a description of the curvature of φ∗ω0 in terms of ρ.
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