This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.
problem Classifying hypersurfaces with parallel Fubini-Pick form in \(\mathbb{R}^{n+1}\).
method Defining a generalized Calabi product and proving decomposition theorems.
result Complete classification of Calabi hypersurfaces in \(\mathbb{R}^{n+1}\) with parallel Fubini-Pick form.
Tian's theorem applies to Moishezon spaces with singular metrics.
problem Distribution of currents on Moishezon spaces with singular metrics.
method Proving asymptotic distribution of Fubini-Study currents.
result Curvature currents of metrics on singular Hermitian line bundles.
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…
New theorem handles stochastic Volterra semimartingales.
problem Classic Fubini theorem restrictions for Volterra semimartingales.
method Introduced measure-valued stochastic integration.
result Proved new stochastic Fubini theorem.
We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on Mn≥3 share the same unparametrized geodesics, and two of them (say, g and gˉ) are strictly nonproportional (i.e., the minimal polynomial of giαgˉαj coincides with …
Extends Euler calculus to continuous integrands using curvature.
problem Limitations of Euler calculus with simple functions.
method Integrates with respect to Gaussian curvature within O-minimal theories.
result Satisfies a Fubini theorem and extends to a functor.
Study shows convergence of Fubini-Study currents to equilibrium metrics on Kähler manifolds.
problem Convergence of Fubini-Study currents to equilibrium metrics in Kähler geometry.
method Analysis of continuous Hermitian metrics and their Fubini-Study currents on line bundles.
result The scaled difference between Fubini-Study currents and equilibrium metrics converges to zero in the sense of currents.
Classifies Calabi hypersurfaces with parallel Fubini-Pick form.
problem Classifying Calabi hypersurfaces with specific geometric properties.
method Classification based on parallel Fubini-Pick form and Levi-Civita connection.
result Classification of 2 and 3-dimensional Calabi hypersurfaces.
Ricci flow simulations show unstable Fubini-Study metrics develop singularities.
problem Understanding the behavior of unstable perturbations in Ricci flow.
method Numerical simulations of Ricci flow starting from unstable Fubini-Study metrics.
result Ricci flow solutions from unstable Fubini-Study metrics develop local singularities.
Proves polynomial injectivity of Fubini-Study map for ample line bundles.
problem Injectivity of Fubini-Study map for ample line bundles.
method Polynomial injectivity proof with polynomial dependence on ample line bundle exponent.
result Quantitative version of injectivity proved, polynomial in ample line bundle exponent.
Rigidity of Fubini-Study metric on odd complex Grassmannians.
problem Characterize infinitesimal deformations of the Fubini-Study metric on complex Grassmannians.
method Explicit description of infinitesimal Einstein deformations, integration analysis.
result Fubini-Study metric on odd complex Grassmannians is rigid.
Paper studies geometric properties of nonlinear Lebesgue spaces.
problem Geometric and analytic properties of nonlinear Lebesgue spaces.
method Formalizes pointwise description of geometric properties using a nonlinear Fubini-Lebesgue theorem.
result Definition of length structure, Alexandrov curvature bounds, and speed for absolutely continuous curves in nonlinear Lebesgue spaces.
In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the class…
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 gFS denotes the Fubini-Study metric of constant hol…
Study of Fubini-Study forms on surfaces with punctures.
problem Analyzing Fubini-Study forms on surfaces with punctures.
method Using Hermitian metrics, holomorphic line bundles, and Kodaira maps.
result Fubini-Study forms grow polynomially near punctures.
We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automatically has contact to order three. We give applications to the study of the Debarre--de Jong conjecture, and of n-dimensional varieties whose…
Link slope stability to Donaldson's functional via Quot-scheme limits.
problem Establishing a connection between slope stability and Donaldson's functional for vector bundles.
method Defining Quot-scheme limits of Fubini-Study metrics and proving Donaldson's functional's coercivity.
result Donaldson's functional is coercive on Fubini-Study metrics for slope stable bundles.
Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.
The Fubini-Study metric minimizes a volume-normalized holomorphic systole in CPn.
problem Finding metrics with minimal holomorphic systoles in complex projective spaces.
method Introduced holomorphic k-systole and used Gauduchon metrics to establish minimization. result The Fubini-Study metric locally minimizes the volume-normalized holomorphic (n−1)-systole. Smooth superspace with special weights has a Fubini-Study form.
problem Describing a new smooth superspace with a special structure.
method Construction of weighted projective superspace and description of its structure.
result Smooth superspace with weights +1,−1 has an analog of the Fubini-Study form. Study extends Nirenberg-Spencer's question to families of submanifolds.
problem Determine the germ of compact complex submanifolds in complex manifolds.
method Reformulate the question for families of submanifolds and their infinitesimal neighborhoods. Prove sufficient conditions for first-order neighborhoods and additional assumptions for submanifolds with nonzero vector fields.
result Affirmative answer to the reformulated question for certain submanifolds.
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.
The paper studies third-order PDEs invariant under affine transformations and connects them to the Fubini-Pick invariant.
problem Investigating third-order PDEs invariant under affine transformations.
method Using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant PDEs on homogeneous manifolds].
result Derives third-order PDEs from the Fubini-Pick invariant.
Study continuity of Bergman kernels on degenerating varieties.
problem Continuity and uniform convergence of Bergman kernels on degenerating varieties.
method Introduced fiberwise Bergman kernel for flat families of polarized varieties, established continuity and uniform convergence results.
result Uniform convergence of Fubini-Study currents and continuity of fiberwise Bergman kernel on test configurations.
New proof of Donaldson-Uhlenbeck-Yau theorem using variational approach.
problem Proving Donaldson-Uhlenbeck-Yau theorem for slope stable holomorphic vector bundles.
method Variational approach, focusing on Bergman kernel asymptotics.
result Elementary proof of Donaldson-Uhlenbeck-Yau theorem with uniform coercivity.
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…
A new geometric metric identifies true data changes from parametrization artifacts in high-dimensional representations.
problem Quantifying representation drift in high-dimensional data using Euclidean or cosine distances can misattribute changes due to arbitrary parametrizations.
method Introducing the Fubini Study metric to identify representations that differ only by gauge transformations.
result The Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations, providing a diagnostic for meaningful structural changes.
Quantum Natural Gradient uses quantum geometry for optimization.
problem Optimizing variational quantum circuits efficiently.
method Quantum generalization of Natural Gradient Descent using Quantum Information Geometry.
result Efficient algorithm for computing metric tensor approximations.
Study on complex spaces and their properties.
problem Characterizing relative properties of complex spaces.
method Analyzing indefinite Euclidean and non-flat complex spaces.
result Indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form.
Killing tensors on complex projective space are identified and generated by Killing fields.
problem Identifying Killing tensors on complex projective space.
method Determining Killing tensors of arbitrary rank on complex projective space with Fubini-Study metric.
result Complex projective spaces are generated by Killing fields.
Maps associate metrics to forms on Kähler manifolds.
problem Mapping metrics to forms on Kähler manifolds.
method Proving any positive hermitian form can be an L2-inner product of a metric, applying to Fubini-Study map.
result Fubini-Study map is injective.
Classifies Kähler-Einstein manifolds with specific properties.
problem Classifying Kähler-Einstein manifolds with rotation invariant metrics.
method Uses Kähler immersions into complex projective spaces with Fubini-Study metrics.
result Classifies manifolds with codimension ≤ 3 and rotation invariant metrics.
We classify all Kahler metrics in an open subset of C2 whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on CP2 restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the E…
We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal…
We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the first step we consider the locally strongly convex ones. In fact, by the idea used by Naitoh, we provide in this paper a direct proof of the complete classification for those affine symmetric hyperspheres. Then, via an earlier result of the fir…
The paper studies Fubini-Study metrics and zero distributions on CR manifolds.
problem Understanding Fubini-Study metrics on CR manifolds.
method Asymptotic analysis of Toeplitz operators and pull-back metrics.
result Established the distribution of zero divisors of random CR functions.
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 ≥3 unless it is essentially the Fubini-Study metric, or the h-projective equivalence is a…
New framework for logarithmically divergent integrals on manifolds with corners.
problem Logarithmically divergent integrals on manifolds with corners.
method Introduces new geometric framework and morphisms in logarithmic geometry.
result Functorial characterization of regularized integration.
Compact Kähler manifolds with specific curvature properties are rigid.
problem Diameter rigidity of Kähler manifolds with positive bisectional curvature.
method Proof of biholomorphic isometry under specific diameter and volume conditions.
result Compact Kähler manifolds with the specified properties are biholomorphically isometric to CPn. We prove a general extrinsic rigidity theorem for homogeneous varieties in CPN. The theorem is used to show that the adjoint variety of a complex simple Lie algebra g (the unique minimal G orbit in Pg) is extrinsically rigid to third order. In contrast, we show that the ad…
Listed Kaehler-Einstein manifolds in complex projective spaces.
problem Classical problem of finding non-isometric Kaehler-Einstein manifolds.
method Complete list for n≤6 using Tn-invariant manifolds. result Solved the problem for n≤6. Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.
problem Existence of balanced metrics and Gieseker stability of vector bundles.
method Quot-scheme limit of Fubini-Study metrics and Bergman 1-parameter subgroups.
result Existence of balanced metrics equivalent to Gieseker stability for singular cases.
The paper defines conditional nonlinear expectations for specific function spaces.
problem Defining conditional nonlinear expectations for specific function spaces.
method Develops a sublinear increasing functional and a set-valued mapping to represent conditional expectations.
result Characterizes the existence and properties of conditional nonlinear expectations.
Geometric quantization results for Riemann surfaces with semi-positive line bundles.
problem Analyzing geometric quantization for Riemann surfaces with semi-positive line bundles.
method Exploring the Bergman kernel expansion and related results for induced Fubini-Study metrics, Toeplitz operators, and holomorphic torsion.
result Asymptotic results for holomorphic torsion and random sections.
Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-o…
The Bergman kernels of holomorphic vector bundles are studied to extend the Fubini-Study map.
problem Extending the Fubini-Study map to a closed range for general inner products.
method Associate Bergman kernels with general inner products on the dual space.
result FS is an injective immersion but not necessarily closed in the space of positive definite inner products.
Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.
problem Classifying self-dual almost-Kähler four-manifolds.
method Using LeBrun's result and properties of Ricci tensor, the authors classify manifolds of different types.
result Any self-dual almost-Kähler metric on CP2 is the Fubini-Study metric up to rescaling. Quantizes geodesics in Kähler and Sasaki geometry.
problem Quantize geodesics in Kähler and Sasaki spaces.
method Classical Fubini-Study map and quantization procedure.
result Proves conditions for geodesics in Kähler potentials.