Fabio Zuddas

Scalar curvature and asymptotic Chow stability of projective bundles and blowups

The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme of P n . These invariants are calculated in two special cases. The first is a projective bundle P(E) over a curve of genus g � 2, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the bundle E is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarization is sufficiently divisible. Moreover it implies that P(E) is asymptotically Chow polystable (with every polarization) if and only if it admits a constant scalar curvature Kahler metric. The second case is a manifold blown-up at points, and new examples of asymptotically Chow unstable constant scalar curvature Kahler classes are given.

Symplectic capacities of Hermitian symmetric spaces of compact and noncompact type

Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer--Zehnder capacity for Cartan domains and their products.

Some remarks on the Gromov width of homogeneous Hodge manifolds

We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds (M, ω) with b2(M) = 1. As an application we obtain an upper bound on the Seshadri constant ϵ(L) where L is the ample line bundle on M such that .

RIEMANNIAN GEOMETRY OF HARTOGS DOMAINS

Let DF = {(z0, z) ∈ ℂn | |z0|2 < b, ||z||2 < F(|z0|2)} be a strongly pseudoconvex Hartogs domain endowed with the Kahler metric gF associated to the Kahler form . This paper contains several results on the Riemannian geometry of these domains. These are summarized in Theorems 1.1–1.3. In the first one we prove that if DF admits a non-special geodesic (see definition below) through the origin whose trace is a straight line then DF is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of DF do not self-intersect, we find necessary and sufficient conditions on F for DF to be geodesically complete and we prove that DF is locally irreducible as a Riemannian manifold. Finally, in Theorem 1.3, we compare the Bergman metric gB and the metric gF in a bounded Hartogs domain and we prove that if gB is a multiple of gF, namely gB = λ gF, for some λ ∈ ℝ+, then DF is holomorphically isometric to an open subset of the complex hyperbolic space.

Canonical metrics on Hartogs domains

An

ENGLIŠ EXPANSION FOR HARTOGS DOMAINS

n

The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold

-dimensional Hartogs domain

Bochner Coordinates on Flag Manifolds

D_F

On the third coefficient of TYZ expansion for radial scalar flat metrics

with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric

Explicit Global Symplectic Coordinates on Kähler Manifolds

g_F