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Andrea Galasso

Publications and source records attributed to Andrea Galasso.

13 recordsLinked to original sources
arXiv2026

Pseudo--K\"ahler induction on Lie groups with entire Grauert tubes

Given a closed subgroup $H\subseteq G$ and a Hamiltonian $H$-space $Y$, one can construct an induced Hamiltonian $G$-space. In this paper we investigate this construction in the setting where $Y$ is endowed with a compatible complex structure. Our aim is to develop symplectic induction in this framework and to establish the corresponding K\"ahler analogues of the classical results. The main idea is to replace the cotangent bundle $T^*G$, appearing in the standard construction of $\operatorname{Ind}$, with the Grauert tube of $G$. We focus on Lie groups admitting a globally defined Grauert tube and study the resulting pseudo--K\"ahler induction procedure.

math.SG↗
arXiv2024

Quantization and reduction for torsion free CR manifolds

Consider a compact torsion free CR manifold $X$ and assume that $X$ admits a compact CR Lie group action $G$. Let $L$ be a $G$-equivariant rigid CR line bundle over $X$. It seems natural to consider the space of $G$-invariant CR sections in the high tensor powers as quantization space, on which a certain weighted $G$-invariant Fourier-Szeg\H{o} operator projects. Under certain natural assumptions, we show that the group invariant Fourier-Szeg\H{o} projector admits a full asymptotic expansion. As an application, if the tensor power of the line bundle is large enough, we prove that quantization commutes with reduction.

math.CV↗
arXiv2024

Strict Quantization for Compact Pseudo-K\"ahler Manifolds and Group Actions

The asymptotic results for Berezin-Toeplitz operators yield a strict quantization for the algebra of smooth functions on a given Hodge manifold. It seems natural to generalize this picture for quantizable pseudo-K\"ahler manifolds in presence of a group action. Thus, in this setting we introduce a Berezin transform which has a complete asymptotic expansion on the preimage of the zero set of the moment map. It leads in a natural way to prove that certain quantization maps are strict.

math.SG↗
arXiv2024

Commutativity of quantization with conic reduction for torus actions on compact CR manifolds

We define conic reduction $X^{\mathrm{red}}_ν$ for torus actions on the boundary $X$ of a strictly pseudo-convex domain and for a given weight $ν$ labeling a unitary irreducible representation. There is a natural residual circle action on $X^{\mathrm{red}}_ν$. We have two natural decompositions of the corresponding Hardy spaces $H(X)$ and $H(X^{\mathrm{red}}_ν)$. The first one is given by the ladder of isotypes $H(X)_{kν}$, $k\in\mathbb{Z}$, the second one is given by the $k$-th Fourier components $H(X^{\mathrm{red}}_ν)_k$ induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for $k$ sufficiently large. The result is given for spaces of $(0,q)$-forms with $L^2$-coefficient when $X$ is a CR manifold with non-degenerate Levi-curvature.

math.CV↗
arXiv2024

Remarks on asymptotic isometric embeddings of conic transforms for torus actions

Consider a Hodge manifold and assume that a torus acts on it in a Hamiltonian and holomorphic manner and that this action linearizes on a given quantizing line bundle. Inside the dual of the line bundle one can define the circle bundle, which is a strictly pseudoconvex CR manifold. Then, there is an associated unitary representation on the Hardy space of the circle bundle. Under suitable assumptions on the moment map, we consider certain loci in unit circle bundle, naturally associated to a ray through an irreducible weight. Their quotients are called conic transforms. We introduce maps which are asymptotic embeddings of conic transforms making use of the corresponding equivariant Szegő projector.

math.DG↗
arXiv2022

Embedding theorems for quantizable pseudo-Kähler manifolds

Given a compact quantizable pseudo-Kähler manifold $(M,ω)$ of constant signature, there exists a Hermitian line bundle $(L,h)$ over $M$ with curvature $-2πi\,ω$. We shall show that the asymptotic expansion of the Bergman kernels for $L^{\otimes k}$-valued $(0,q)$-forms implies more or less immediately a number of analogues of well-known results, such as Kodaira embedding theorem and Tian's almost-isometry theorem.

math.DG↗
arXiv2022

Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

Given a CR manifold with non-degenerate Levi form, we show that the operators of the functional calculus for Toeplitz operators are complex Fourier integral operators of Szegő type. As an application, we establish semi-classical spectral dimensions for Toeplitz operators. We then consider a CR manifold with a compact Lie group action $G$ and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also compute semi-classical spectral dimensions for $G$-invariant Toeplitz operators.

math.FA↗
arXiv2022

On the singularities of the Szegő kernels on CR orbifolds

In this paper we study the microlocal properties of the Szegő kernel of a given compact connected orientable CR orbifold whose Kohn Laplacian has closed range. This last assumption is satisfied if certain geometric conditions hold true, as in the smooth case. As applications, we give a pure analytic proof of Kodaira-Bailey theorem and explain how to generalize a CR version of quantization commutes with reduction to orbifolds.

math.CV↗
arXiv2021

Equivariant asymptotics of Szegö kernels under Hamiltonian $SU(2)\times S^1$-actions

Let $M$ be complex projective manifold and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian and holomorphic manner and that this action linearizes to $A$. Then, there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space $H(X)$. The standard circle action on $H(X)$ commutes with the action of $G$ and thus one has a decompositions labeled by $(k\,\boldsymbolν,\,k)$, where $k\in\mathbb{Z}$ and $\boldsymbol{ ν}\in \hat{G}$. We consider the local and global asymptotic properties of the corresponding equivariant projector as $k$ goes to infinity. More generally, for a compact connected Lie group, we compute the asymptotics of the dimensions of the corresponding isotypes.

math.SG↗
arXiv2021

Toeplitz operators on CR manifolds and group actions

Let $(X, T^{1,0}X)$ be a connected orientable compact CR manifold of dimension $2n+1$, $n \geq 1$ with non-degenerate Levi curvature. In this paper, we study the algebra of Toeplitz operators on $X$ and we establish star product for some class of symbols on $X$. In the second part of this paper, we consider a compact locally free Lie group $G$ acting on $X$ and we investigate the associated algebra of $G$-invariant Toeplitz operators.

math.CV↗
arXiv2018

Equivariant Asymptotics of Szegö kernels under Hamiltonian $SU(2)$-action

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that $SU(2)$ acts on $M$ in a Hamiltonian manner, with nowhere vanishing moment map, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space, and the isotypical components are all finite dimensional. We consider the local and global asymptotic properties the equivariant projector associated to a weight $k \, \boldsymbol{ ν}$, when $\boldsymbol{ ν}$ is fixed and $k\rightarrow +\infty$.

math.SG↗
arXiv2018

Equivariant Asymptotics of Szegö kernels under Hamiltonian $U(2)$ actions

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian manner, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space. If the moment map is nowhere vanishing, the isotypical component are all finite dimensional, they are generally not spaces of sections of some power of $A$. One is then led to study the local and global asymptotic properties the isotypical component associated to a weight $k \, \boldsymbolν$, when $k\rightarrow +\infty$. In this paper, part of a series dedicated to this general theme, we consider the case $G=U(2)$.

math.SG↗