Narrow Results

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\ge 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle ${\mathrm{\Omega }}_X^{\otimes k}$, where ${\mathrm{\Omega }}_X$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding Bergman metric on $X$ in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of $X$ into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of ${\mathrm{\Omega }}_X^{\otimes k}$. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of $X$. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of ${\mathrm{\Omega }}_X^{\otimes k}$ and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on $X$.

In this paper we investigate the quantum cohomologies of symmetric products of Kähler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov–Witten invariant and relative Gromov–Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line ℙ¹, which is the k-dimensional complex projective space ℙ^k.

The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A novel geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpretation of the symmetric product yields two different applications. First, an intrinsic proof is provided of the fact that the distributions closed under the symmetric product are exactly those distributions invariant under the geodesic flow. Second, some applications in geometric control theory for mechanical systems are clarified.

Let $X$ be a compact connected Riemann surface of genus at least two. The Abel–Jacobi map $\phi :{\mathrm{\text{Sym}}}^d(X)\to {\mathrm{\text{Pic}}}^d(X)$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the pull-back, by $\phi$, of the flat metric on ${\mathrm{\text{Pic}}}^d(X)$. 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.