Abel-Jacobi map and curvature of the pulled back metric

Let $X$ be a compact connected Riemann surface of genus at least two. The Abel-Jacobi map $\varphi: {\rm Sym}^d(X) \rightarrow {\rm Pic}^d(X)$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the pull-back, by $\varphi$, of the flat metric on ${\rm Pic}^d(X)$. In particular, we show that when $d=1$, the curvature is strictly negative everywhere if $X$ is not hyperelliptic, and when $X$ is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of $X$ fixed by the hyperelliptic involution.


Introduction
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 γ X such that there is a nonconstant holomorphic map from X to CP 1 of degree γ X . Consider the Abel-Jacobi map ϕ : Sym d (X) −→ Pic d (X) that sends an effective divisor D on X of degree d to the corresponding holomorphic line bundle O X (D). If d < γ X , then ϕ is an embedding (Lemma 2.1). On the other hand, Pic d (X) is equipped with a flat Kähler form, which we will denote by ω 0 . So, ϕ * ω 0 is a Kähler form on Sym d (X), whenever d < γ X . The Kähler metric ϕ * ω 0 on Sym d (X) is relevant in the study of abelian vortices (see [Ri], [BR] and references therein).
Our aim here is to study the curvature of this Kähler form ϕ * ω 0 on Sym d (X).
Consider the g-dimensional vector space H 0 (X, K X ) consisting of holomorphic oneforms on X. It is equipped with a natural Hermitian structure. Let be the Grassmannian parametrizing all d dimensional quotients of H 0 (X, K X ). The Hermitian structure on H 0 (X, K X ) produces a Hermitian structure on the tautological vector bundle on G of rank d; this tautological bundle on G of rank d will be denoted by V . The Hermitian structure on H 0 (X, K X ) also gives a Fubini-Study Kähler form on G.
There is a natural holomorphic map ρ : Sym d (X) −→ G (see (3.6)). We prove that the holomorphic Hermitian vector bundle ρ * V −→ Sym d (X) is isomorphic to the holomorphic cotangent bundle Ω Sym d (X) equipped with the Hermitian structure given by ϕ * ω 0 (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 * ω 0 in terms of ρ. In particular, we show that when d = 1, the curvature of ϕ * ω 0 is strictly negative if X is not hyperelliptic; on the other hand, if X is hyperelliptic, then the curvature of ϕ * ω 0 vanishes at the 2(g +1) points of X fixed by the hyperelliptic involution; the curvature of ϕ * ω 0 is strictly negative outside these 2(g + 1) points (Proposition 3.2).

Gonality and flat metric
As before, X is a compact connected Riemann surface of genus g, with g ≥ 2. For any positive integer d, let Sym d (X) denote the quotient of the Cartesian product X d under the natural action of the group of permutations of {1, · · · , n}. This Sym d (X) is a smooth complex projective manifold of dimension d. The component of the Picard group of X parametrizing the holomorphic line bundles of degree d will be denoted by Pic d (X).
This map ϕ is surjective if and only if d ≥ g.
Consider the space of all nonconstant holomorphic maps from X to the complex projective line CP 1 . More precisely, consider the degree of all such maps. Since g ≥ 2, the degree of any such map is at least two. The gonality of X is defined to the smallest integer among the degrees of maps in this space [Ei,p. 171]. Equivalently, the gonality of X is the smallest one among the degrees of holomorphic line bundles L on X with dim H 0 (X, L) ≥ 2. The gonality of X will be denoted by γ X . Note that γ X = 2 if and only if X is hyperelliptic. The gonality of a generic compact Riemann surface of genus g is g+3 2 . Lemma 2.1. Assume that d < γ X . Then the map ϕ in (2.1) is an embedding.
Proof. We will first show that ϕ is injective.

Take any point
there is a meromorphic function on X with pole divisor D y and zero divisor D x . In particular, the degree of this meromorphic function is d. But this contradicts the given condition that d < γ X . Consequently, the map ϕ is injective.
We need to show that for x ∈ Sym d (X), the differential is injective.
We will quickly recall a description of the tangent bundle T Sym d (X).
be the tautological reduced effective divisor consisting of all ({y 1 , · · · , y d }, y) such that y ∈ {y 1 , · · · , y d }. The projection of Sym d (X) × X to Sym d (X) will be denoted by p.
Consider the quotient sheaf Note that its support is the divisor D. The tangent bundle T Sym d (X) is the direct image Take any be the short exact sequence of sheaves on X, where D x , as before, is the effective divisor given by x. Tensoring the sequence in ( we obtain the following short exact sequence of sheaves on X: be the long exact sequence of cohomologies associated to the short exact sequence of sheaves in (2.5). From the earlier description of T Sym d (X) we have the following: Hence the homomorphism α in (2.5) is an isomorphism. This implies that the homomorphism δ x in (2.5) is injective. So the exact sequence in (2.5) gives the exact sequence The tangent bundle of Pic d (X) is the trivial vector bundle over Pic d (X) with fiber H 1 (X, O X ). The differential dϕ(x) in (2.2) coincides with the homomorphism δ x in (2.5).
Since δ x in (2.8) is injective, it follows that dϕ(x) is injective.
Let K X denote the holomorphic cotangent bundle of X. The vector space H 0 (X, K X ) is equipped with the Hermitian form θ 1 , θ 2 := X θ 1 ∧ θ 2 ∈ C , θ 1 , θ 2 ∈ H 0 (X, K X ) . (2.9) This Hermitian form on H 0 (X, K X ) produces a Hermitian form on the dual vector space this isomorphism is given by Serre duality. This Hermitian form on H 1 (X, O X ) produces a Kähler structure on Pic d (X) which is invariant under the translation action of Pic 0 (X) on Pic d (X). This Kähler structure on Pic d (X) will be denoted by ω 0 . Now Lemma 2.1 has the following corollary: Corollary 2.2. Assume that d < γ X . Then ϕ * ω 0 is a Kähler structure on Sym d (X).

Mapping to a Grassmannian
We will always assume that d < γ X . Since γ X ≤ g, we have d < g.
be the Grassmannian parametrizing all d dimensional quotients of H 0 (X, K X ). Let be the tautological vector bundle of rank d. So V is a quotient of the trivial vector bundle G × H 0 (X, K X ) −→ G. Consider the Hermitian form on H 0 (X, K X ) defined in (2.9). It produces a Hermitian structure on the trivial vector bundle G × H 0 (X, K X ) −→ G.
Identifying the quotient V with the orthogonal complement of the kernel of the projection to V , we get a Hermitian structure on V . Let be this Hermitian structure on V .
Take any x := {x 1 , · · · , x d } ∈ Sym d (X). Consider the short exact sequence of sheaves on X where D x as before is the divisor on X given by x. Let be the long exact sequence of cohomologies associated to it. By Serre duality, hence from (2.7) it follows that α ′ in (3.5) is an isomorphism. This implies that β ′ in (3.5) is the zero homomorphism, hence δ ′ x is surjective. In other words, H 0 (X, Q ′ (x)) is a quotient of H 0 (X, K X ) of dimension d. Therefore, H 0 (X, Q ′ (x)) gives a point of the Grassmannian G constructed in (3.1).
Proof. We will show that the homomorphisms α ′ , β ′ , δ ′ x and ν ′ in (3.5) are duals of the homomorphisms α, β, δ x and ν respectively, which are constructed in (2.5). This actually follows from the fact that the complex in (3.4) is dual of the complex in (2.4). We will elaborate this a bit.
The isomorphism H 1 (X, O X ) * = H 0 (X, K X ) in (3.7) is an isometry, because the Hermitian form on H 1 (X, O X ) * is defined using the Hermitian form on H 0 (X, K X ) in (2.9) and this isomorphism. This implies that the isomorphism in (3.8) is an isometry, after H 0 (X, Q(x)) (respectively, H 0 (X, Q ′ (x))) is equipped with the Hermitian structure obtained from the Hermitian structure on H 1 (X, O X ) (respectively, H 0 (X, K X )) using (2.5) (respectively, (3.5)). This completes the proof.
(1) If X is not hyperelliptic, then Θ is strictly negative everywhere on X.