International Journal of Mathematics

Let X be the wonderful compactification of a complex symmetric space G/H of minimal rank. For a point x ∈ G, denote by Z the closure of BxH/H in X, where B is a Borel subgroup of G. The universal cover of G is denoted by . Given a equivariant vector bundle E on X, we prove that E is nef (respectively, ample) if and only if its restriction to Z is nef (respectively, ample). Similarly, E is trivial if and only if its restriction to Z is so.

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold equipped with the metric g = dr² + λ(r)² g_N. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)^αp_k, or constant weighted higher-order mean curvature ratio (λ′)^αp_k/p₁, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here p_k is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝⁿ, ℍⁿ and , which generalize the classical Alexandrov-type results.

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,…,L〉 → S with the determinant line bundle (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen–Mumford expansion, J. Differential Geom. 78 (2008) 475–496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles and 〈L,…,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,…,L〉 → S and is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91–96] between a Deligne paring and a certain determinant line bundle.

We prove that the group of contact diffeomorphisms is closed in the group of all diffeomorphisms with respect to the C⁰-topology. By Gromov's alternative, it suffices to exhibit a diffeomorphism that cannot be approximated uniformly by contact diffeomorphisms. Our construction uses Eliashberg's shape invariant.

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, ℂ). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.