arXiv research

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

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

Rigidity of Fubini-Study metric on odd complex Grassmannians.

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

Study of Fubini-Study forms on surfaces with punctures.

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…

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

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

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

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.

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

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…

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

Listed Kaehler-Einstein manifolds in complex projective spaces.

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

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…

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…

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.

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 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 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 $(2m(m+1))^{1/2}$.

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…

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…

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.

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

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

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

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.

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.

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…

The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for…

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

SSDMs generate quantum states directly, outperforming classical methods.

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.

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.

Solves classical problem with Kähler-Einstein metrics in complex projective spaces.

In \cite{D3}, Donaldson defines a dynamical system on the space of Fubini-Study metrics on a polarized compact Kähler manifold. Sano proved that if there exists a balanced metric for the polarization, then this dynamical system always converges to the balanced metric (\cite{S}). In \cite{DKLR}, Douglas, et. al., conjec…

Uniform convergence of metrics on vortex moduli space in Bradlow limit.

Study classifies Kähler-Einstein metrics with rotational symmetries.

We present three families of exact, cohomogeneity-one Einstein metrics in $(2n+2)$ dimensions, which are generalizations of the Stenzel construction of Ricci-flat metrics to those with a positive cosmological constant. The first family of solutions are Fubini-Study metrics on the complex projective spaces $CP^{n+1}$, w…

We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the Laplace spectrum. In particular, we prove dynamical stability of some classes of E…

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

We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This …

Considering a non-constant smooth solution $f$ of the Tanno equation on a closed, connected Kähler manifold $(M,g,J)$ with positively definite metric $g$, Tanno showed that the manifold can be finitely covered by $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS})$, where $g_{FS}$ denotes the Fubini-Study metric of constant hol…

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

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…

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini--Study metric. This conjecture is verified in all odd dimensions, for …

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.