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A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed in complex dimension $2$, by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to the class of complex nilmanifolds and confirm the conjecture in that case.

In this letter, we focus on numerical solutions to the problem of quantizing circularly symmetric complex Gaussian random variables from open symmetrized bidiscs in $\mathbb {C}^{2}$ . To that end, we leverage techniques from the complex variable analysis, especially the symmetrization map. Four novel non-uniform quantizers, i.e. Bd1/2-GQ and EC Bd1/2-GQ, are proposed to reduce the mean square error distortion. Compared to the existing high-rate golden quantizer, entropy-coded high-rate golden quantizer, and (non)-uniform polar/rectangular quantizers, Bd1-GQ and EC Bd1-GQ achieve lower mean square error distortion. Moreover, the EC Bd2-GQ reaches better performance than the polar- and rectangular- quantizers.

Probability measures with either finite Monge-Amp\`ere energy or finite entropy have played a central role in recent developments in K\"ahler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge-Amp\`ere measures have finite entropy. We show that these potentials belong to the finite energy class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions., Comment: 22 pages

On a complete noncompact Kahler manifold $$M^{n}$$ (complex dimension) with nonnegative Ricci curvature and Euclidean volume growth, we prove that polynomial growth holomorphic functions of degree d has an dimension upper bound $$cd^{n}$$ , where c depends only on n and the asymptotic volume ratio. Note that the power is sharp.

The Torelli group $$\mathcal T(X)$$ T ( X ) of a closed smooth manifold X is the subgroup of the mapping class group $$\pi _0(\mathrm {Diff}^+(X))$$ π 0 ( Diff + ( X ) ) consisting of elements which act trivially on the integral cohomology of X. In this note we give counterexamples to Theorem 3.4 by Verbitsky (Duke Math J 162(15):2929–2986, 2013) which states that the Torelli group of simply connected Kähler manifolds of complex dimension $$\ge 3$$ ≥ 3 is finite. This is done by constructing under some mild conditions homomorphisms $$J: \mathcal T(X) \rightarrow H^3(X;\mathbb Q)$$ J : T ( X ) → H 3 ( X ; Q ) and showing that for certain Kähler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in (Verbitsky in Duke Math J 162(15):2929–2986, 2013) where Verbitsky claims that the Torelli group of hyperkähler manifolds are finite. These examples are detected by the action of diffeomorphsims on $$\pi _4(X)$$ π 4 ( X ) . Finally we confirm the finiteness result for the special case of the hyperkähler manifold $$K^{[2]}$$ K [ 2 ] .

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on (m, 0) forms, i.e., sections of the canonical bundle of Kahler manifolds, where m is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of (m, 0) forms, which involves only the Ricci curvature and the gradient of the scalar curvature.

We study the stability of compact pseudo-Kahler manifolds, i.e. compact complex manifolds X endowed with a symplectic form compatible with the complex structure of X. When the corresponding metric is positive-definite, X is Kahler and any sufficiently small deformation of X admits a Kahler metric by a well-known result of Kodaira and Spencer. We prove that compact pseudo-Kahler surfaces are also stable, but we show that stability fails in every complex dimension n ≥ 3 . Similar results are obtained for compact neutral Kahler and neutral Calabi-Yau manifolds. Finally, motivated by a question of Streets and Tian in the positive-definite case, we construct compact complex manifolds with pseudo-Hermitian-symplectic structures that do not admit any pseudo-Kahler metric.

To any completely integrable second-order system of real or complex partial differential equations: $$\begin{aligned} y_{x^{k_1}x^{k_2}} = F_{k_1,k_2} \left( x^1,\dots ,x^n,\,y,\,y_{x^1},\ldots ,y_{x^n} \right) \end{aligned}$$ with $$1 \leqslant k_1,\, k_2 \leqslant n$$ and with $$F_{ k_1, k_2} = F_{ k_2, k_1}$$ in $$\underline{n \geqslant 2}$$ independent variables $$(x^1, \ldots , x^n)$$ and in one dependent variable y, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials to the associated submanifold of solutions, what the vanishing of the Hachtroudi curvature gives can be precisely translated to characterize when both families of Segre varieties and of conjugate Segre varieties associated to a Levi nondegenerate real analytic hypersurface M in $$\mathbb {C}^n$$ ( $$n \geqslant 3$$ ) can be straightened to be affine complex (conjugate) lines. In continuation to a previous paper devoted to the quite distinct $$\mathbb {C}^2$$ -case, this then characterizes in an effective way those hypersurfaces of $$\mathbb {C}^{n+1}$$ in higher complex dimension $$n + 1\geqslant 3$$ that are locally biholomorphic to a piece of the $$(2n + 1)$$ -dimensional Heisenberg quadric, without any special assumption on their defining equations.

Inspired by the pioneer work of H.L. Resnikoff, which is described in full detail in the first part of this two-part paper, we give a quantum description of the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{P}$\end{document}P of perceived colors. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{P}$\end{document}P is the effect space of a rebit, a real quantum qubit, whose state space is isometric to Klein’s hyperbolic disk. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with Hering’s disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy.

In this paper, we study the prescribing Webster scalar curvature problem on strictly pseudoconvex CR manifolds of real dimension $$2n+1$$ . First, we study the CR Nirenberg problem and prove some existence results. Second, we provide the existence and a positive lower bound for a solution of the CR Yamabe problem with zero Webster scalar curvature on noncompact complete manifolds.

Small changes in sections 4 and 5, results not affected, Comment: 17 pages

We introduce the concept of a holomorphic field theory on any complex manifold in the language of the Batalin-Vilkovisky formalism. When the complex dimension is one, this setting agrees with that of chiral conformal field theory. Our main result concerns the behavior of holomorphic theories under renormalization group flow. Namely, we show that holomorphic theories are one-loop finite. We use this to completely characterize holomorphic anomalies in any dimension. Throughout, we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories.

This paper proposed a canny edge detection-based approach for measuring the dimension of an object of mobile type captured in an image from the given distance. The purpose of the scheme is to implement aspects of forensic investigation helpful for our society. It makes use of mobile dimension detection algorithms to narrow it down to a range of mobile phones in a grievous event. The scheme not only identifying the mistrusting individual but also assists the forensics investigation team. In addition to that, the technique support to track the suspected individual as well as also concerns for tapping the communication records of concern person. It has been tested on wide varieties of mobile images of different model from same or different companies. The superiority of the proposed scheme is that excellent results are obtained while comparing the real dimension of an object with the object in an image.