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Roberto Mossa

Publications and source records attributed to Roberto Mossa.

29 records · Page 2Linked to original sources
arXiv2015

Some remarks on homogeneous Kähler manifolds

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous Kähler manifold M endowed with an integral Kähler form $μω$, admits a holomorphic isometric immersion in the complex projective space, for a suitable $μ>0$. This result has two corollaries which extend to homogeneous Kähler manifolds the results obtained by the authors in [8] and in [12] for homogeneous bounded domains.

math.DG↗
arXiv2014

A note on diastatic entropy and balanced metrics

We give un upper bound Ent(Ω, g)<λ of the diastatic entropy Ent(Ω, g) of a complex bounded domain (Ω, g) in terms of the balanced condition (in Donaldson terminology) of the Kaehler metric λg. When (Ω, g) is a homogeneous bounded domain we show that the converse holds true, namely if Ent(Ω, g)<1 then g is balanced. Moreover, we explcit compute Ent(Ω, g) in terms of Piatetski-Shapiro constants.

math.DG↗
arXiv2013

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 $b_2(M)=1$. 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 $c_1(L)=[\fracωπ]$.

math.SG↗
arXiv2013

Symplectic capacities of Hermitian symmetric spaces

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.

math.SG↗
arXiv2012

A bounded homogeneous domain and a projective manifold are not relatives

Let $M_1$ and $M_2$ be two Kähler manifolds. One says that $M_1$ and $M_2$ are "relatives" if they share a non-trivial Kähler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) $h_1: S -> M_1$ and $h_2: S -> M_2$. In this paper we show that a bounded homogeneous domain with a homogeneous Kähler metric and a projective Kähler manifold (i.e. a projective manifold endowed with the restriction of the Fubini-Study metric) are not relatives. Our result is a generalization of the result obtained by A. J. Di Scala and A. Loi (in "Kähler manifolds and their relatives", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 3 (2010), 495-501) for the Bergman metrics.

math.DG↗
arXiv2011

Balanced metrics on homogeneous vector bundles

Let $E\rightarrow M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$ and let $E=E_1\oplus... \oplus E_m\rightarrow M$ be its decomposition into irreducible factors. Suppose that each $E_j$ admits a $ω$-balanced metric in Donaldson-Wang terminology. In this paper we prove that $E$ admits a unique $ω$-balanced metric if and only if $\frac{r_j}{N_j}=\frac{r_k}{N_k}$ for all $j, k=1, ..., m$, where $r_j$ denotes the rank of $E_j$ and $N_j=\dim H^0(M, E_j)$. We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety $(M, ω)$ and we show the existence and rigidity of balanced Kaehler embedding from $(M, ω)$ into Grassmannians.

math.AG↗
arXiv2010

Uniqueness of balanced metrics on holomorphic vector bundles

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$. We prove that if $E$ admits a $ω$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $ω$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of $ω$-balanced Kaehler maps into Grassmannians.

math.DG↗
arXiv2009

The diastatic exponential of a symmetric space

Let $(M, g)$ be a real analytic Kaehler manifold. We say that a smooth map $E_p:W\to M$ from a neighborhood $W$ of the origin of $T_pM$ into $M$ is a {\em diastatic exponential} at $p$ if it satisfies $$(d \E_p)_0=\id_{T_pM},$$ $$D_p(\E_p (v))=g_p(v, v), \forall v\in W,$$ where $D_p$ is Calabi's diastasis function at $p$ (the usual exponential $\exp_p$ obviously satisfied these equations when $D_p$ is replaced by the square of the geodesics distance $d^2_p$ from $p$). In this paper we prove that for every point $p$ of an Hermitian symmetric space of noncompact type M there exists a globally defined diastatic exponential centered in $p$ which is a diffeomorphism and it is uniquely determined by its restriction to polydisks. An analogous result holds true in an open dense neighborhood of every point of $M^*$, the compact dual of $M$. We also provide a geometric interpretation of the symplectic duality map in terms of diastatic exponentials. As a byproduct of our analysis we show that the symplectic duality map pulls back the reproducing kernel of $M^*$ to the reproducing kernel of $M$.

math.SG↗