A complete characterization of bounded and unbounded norm hermitian operators on $H_E^{\infty }$ is given for the case when E is a complex Banach space with trivial multiplier algebra. As a consequence, the bi-circular projections on $H_E^{\infty }$ are determined. We also characterize a subclass of hermitian operators on $S_{}^{\infty }$ for a complex Hilbert space.

To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice...

Let $(X,\omega )$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X,\omega )$ of the complex Monge-Ampère operator acting on $\omega$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^p$-density belong to $(X,\omega )$ and proving that...

The object of the present paper is to study $\eta$-Ricci solitons on $\eta$-Einstein ${\left(LCS\right)}_n$-manifolds. It is shown that if $\xi$ is a recurrent torse forming $\eta$-Ricci soliton on an $\eta$-Einstein ${\left(LCS\right)}_n$-manifold then $\xi$ is (i) concurrent and (ii) Killing vector field.

We present a large class of homogeneous 2-nondegenerate CR-manifolds $M$, both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains $U$, $V$ in $M$ extends to a global real-analytic CR-automorphism of $M$. We show that this class contains $G$-orbits in Hermitian symmetric spaces $Z$ of compact type, where $G$ is a real form of the complex Lie group $Aut{\left(Z\right)}^0$ and $G$ has an open orbit that is a bounded symmetric domain of tube type. ...

Let ${\overline{L}}_i\to X_i$ be a holomorphic line bundle over a compact complex manifold for $i=1,2$. Let $S_i$ denote the associated principal circle-bundle with respect to some hermitian inner product on ${\overline{L}}_i$. We construct complex structures on $S=S_1\times S_2$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that ${\overline{L}}_i$ are equivariant ${\left({ℂ}^{*}\right)}^{n_i}$-bundles satisfying some additional conditions....

We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic $K$-correspondences. We define an intrinsic logarithmic pseudo-volume form ${\Phi }_{X,D}$ for every pair $(X,D)$ consisting of a complex manifold $X$ and a normal crossing Weil divisor $D$ on $X$, the positive part of which is reduced. We then prove that ${\Phi }_{X,D}$ is generically non-degenerate when $X$ is projective...

We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,4}$ to compute the generators of the ${\mathbb{Z}}_2$–cohomology groups $H^j\left({\tilde{G}}_{n,4}\right)$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n,3}$ we conjecture some predictions.

In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma \in {\mathbb{Z}}^{+}$, i.e., $${\Omega }_n\left(\gamma \right)=\{z=(z_1,\cdots ,z_n)\in {ℂ}^n:|z_1{|}^{1/\gamma }<|z_2|<\cdots <|z_n|<1\}$$ equipped with a natural Kähler form ${\omega }_{g\left(\mu \right)}:=\frac{1}{2}(i/\pi )\partial \overline{\partial }{\Phi }_n$ with $${\Phi }_n\left(z\right)=-{\mu }_{1}ln\left(\right|z_2{|}^{2\gamma }-\left|z_1{|}^2\right)-\sum _{i=2}^{n-1}{\mu }_{i}ln\left(\right|z_{i+1}{|}^2-\left|z_i{|}^2\right)-{\mu }_{n}ln(1-|z_n{|}^2),$$ where $\mu =({\mu }_1,\cdots ,{\mu }_n)$, ${\mu }_i>0$, depending on $n$ parameters. The purpose of this paper is threefold. First, we compute the explicit expression for the weighted Bergman kernel function for $({\Omega }_n\left(\gamma \right),g\left(\mu \right))$ and we prove that $g\left(\mu \right)$ is balanced if and only if ${\mu }_1>1$ and $\gamma {\mu }_1$ is an integer, ${\mu }_i$ are integers such that ${\mu }_i\ge 2$ for all $i=2,...,n-1$, and ${\mu }_n>1$. Second, we prove that $g\left(\mu \right)$ is Kähler-Einstein if and only if ${\mu }_1={\mu }_2=\cdots ={\mu }_n=2\lambda$, where...

Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

Let $(M,\theta )$ be a compact CR manifold of dimension $2n+1$ with a contact form $\theta$, and $L=(2+2/n){\Delta }_b+R$ its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form $\tilde{\theta }$ on $M$ conformal to $\theta$ which has a constant Webster curvature. This problem is equivalent to the existence of a function $u$ such that $Lu=u^{1+2/n}$, $u>0$ on $M$. D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where $n\ge 2$ and $(M,\theta )$ is not locally CR equivalent to the sphere $S^{2n+1}$ of ${𝐂}^n$. In a join work with R. Yacoub, the CR Yamabe...

Let $-(n+1)<m\le -(n+1)(1-\rho )$ and let $T_a\in {ℒ}_{\rho ,\delta }^m$ be pseudo-differential operators with symbols $a(x,\xi )\in {ℝ}^n\times {ℝ}^n$, where $0<\rho \le 1$, $0\le \delta <1$ and $\delta \le \rho$. Let $\mu$, $\lambda$ be weights in Muckenhoupt classes $A_p$, $\nu ={\left(\mu {\lambda }^{-1}\right)}^{1/p}$ for some $1<p<\infty$. We establish a two-weight inequality for commutators generated by pseudo-differential operators $T_a$ with weighted BMO functions $b\in {\mathrm{BMO}}_{\nu }$, namely, the commutator $[b,T_a]$ is bounded from $L^p\left(\mu \right)$ into $L^p\left(\lambda \right)$. Furthermore, the range of $m$ can be extended to the whole $m\le -(n+1)(1-\rho )$.

Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{𝒥}$ satisfy a certain...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi }_V$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^m$-measurable subset of ℝⁿ with ${}^m\left(A\right)<\infty$. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V,v)\left|V\in G\right(n,m),v\in V$ such that, for all P ∈ A, one has ${}^{m(n-m)}\left(V\in G(n,m)\left|\right(V,{\pi }_V\left(P\right))\in Z\right)>0$. One can replace “for all P ∈ A” by “for...