arXiv research

Proves polynomial injectivity of Fubini-Study map for ample line bundles.

Study of Fubini-Study forms on surfaces with punctures.

Suppose that we have a compact Kähler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We appl…

The Bergman kernels of holomorphic vector bundles are studied to extend the Fubini-Study map.

The paper studies Fubini-Study metrics and zero distributions on CR manifolds.

Quantizes geodesics in Kähler and Sasaki geometry.

Study shows convergence of Fubini-Study currents to equilibrium metrics on Kähler manifolds.

Ricci flow simulations show unstable Fubini-Study metrics develop singularities.

We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.

Rigidity of Fubini-Study metric on odd complex Grassmannians.

The Fubini-Study metric minimizes a volume-normalized holomorphic systole in $\mathbb{C}P^n$.

Smooth superspace with special weights has a Fubini-Study form.

The Madelung transform connects quantum mechanics and hydrodynamics.

We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…

We give a necessary and sufficient condition for a special Lagrangian submanifold in C^n constructed by Lawlor being also special Lagrangian in C^n with the Fubini-Study form.

Study proves rigidity of harmonic maps from 2-torus to complex projective space.

For a holomorphic vector bundle $E$ over a polarised Kähler manifold, we establish a direct link between the slope stability of $E$ and the asymptotic behaviour of Donaldson's functional, by defining the Quot-scheme limit of Fubini-Study metrics. In particular, we provide an explicit estimate which proves that Donaldso…

Study continuity of Bergman kernels on degenerating varieties.

A new geometric metric identifies true data changes from parametrization artifacts in high-dimensional representations.

Tian's theorem applies to Moishezon spaces with singular metrics.

Killing tensors on complex projective space are identified and generated by Killing fields.

The degree of mobility of a (pseudo-Riemannian) Kähler metric is the dimension of the space of metrics h-projectively equivalent to it. We prove that a metric on a closed connected manifold can not have the degree of mobility $\ge 3$ unless it is essentially the Fubini-Study metric, or the h-projective equivalence is a…

Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature fo…

Listed Kaehler-Einstein manifolds in complex projective spaces.

A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of…

Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.

Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.

The Madelung transform is known to relate Schrödinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a Kähler map (i.e. a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to t…

It was shown by Seaman that if a compact, oriented 4-dimensional riemannian manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, its intersection form is definite and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomor…

In this paper we construct and study a new 15-vertex triangulation $X$ of the complex projective plane $\CP^2$. The automorphism group of $X$ is isomorphic to $S_4\times S_3$. We prove that the triangulation $X$ is the minimal by the number of vertices triangulation of $\CP^2$ admitting a chess colouring of four-dimens…

SSDMs generate quantum states directly, outperforming classical methods.

Sharp inequality for complex projective space's systoles under scalar curvature bounds.

We show that an indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form. We further study whether two compact Fubini-Study spaces are relatives or not.

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

We consider complex projective space with its Fubini-Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.

We classify Kähler-Einstein manifolds which admit a Kähler immersion into a finite dimensional complex projective space endowed with the Fubini-Study metric, whose codimention is not greater than 3 and whose metric is rotation invariant.

We prove that the largest first eigenvalue of the Dirac operator among all hermitian metrics on the complex projective space of odd dimension $m$, larger than the Fubini-Study metric is bounded by $(2m(m+1))^{1/2}$.

In this paper we study the smallest non-zero eigenvalue $λ_1$ of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for $λ_1$ in terms of moment polytope data. We show that this bound can only be attained for $\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\mathbb{CP}^n$ endowe…

Let (M,g) be a compact oriented Einstein 4-manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, equipped with its standard Fubini-Study metric.

In this paper we will prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is the complex projective plane CP^2, with its Fubini-Study metric.

We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.

Study of random sections on complex spaces converging to equilibrium metrics.

Max diameter Kahler manifolds with positive bisectional curvature are complex projective spaces.

Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-Kähler 4-manifolds.

Let $M^n$ be a compact Kähler manifold with bisectional curvature bounded from below by $1$. If $diam(M) = π/ \sqrt{2}$ and $vol(M)> vol(\mathbb{C}\mathbb{P}^n)/ 2^n$, we prove that $M$ is biholomorphically isometric to $\mathbb{C}\mathbb{P}^n$ with the standard Fubini-Study metric.

We classify all Kahler metrics in an open subset of $C^2$ whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on $CP^2$ restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the E…

Zoll manifolds with entire Grauert tubes are proven to be standard complex projective spaces.

We show that a compact complex surface which admits a conformally Kähler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g beco…