Proves inequality between Betti numbers and total curvature of complex projective manifolds.
problem Understanding the relationship between curvature and topological invariants of complex projective manifolds.
method Analyzes the sum of Betti numbers and total curvature of complex projective manifolds.
result Characterizes manifolds with minimal total curvature and extends classical theorems.
Characterizes projective special complex manifolds using c-projective structures.
problem Characterizing projective special complex manifolds.
method Defining S1-bundles and constructing conical special complex manifolds. result Intrinsic characterization of projective special complex manifolds.
Every 4-manifold can be smoothly embedded in complex projective 3-space.
problem Embedding 4-manifolds in complex projective spaces.
method Analyzing properties of 4-manifolds and complex projective spaces.
result Every closed orientable smooth 4-manifold admits a smooth embedding in $\CP^3$.
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. Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
Max diameter Kahler manifolds with positive bisectional curvature are complex projective spaces.
problem Understanding the rigidity of Kahler manifolds with specific curvature properties.
method Proving isometry to complex projective space using maximal diameter and positive bisectional curvature.
result Kahler manifolds with maximal diameter and positive bisectional curvature are isometric to complex projective spaces.
Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.
problem Characterizing positively curved manifolds with torus symmetry.
method Analyzing actions of 3-dimensional tori on closed, simply connected 10-manifolds. result Closed, simply connected, positively curved 10-manifolds with T3-symmetry are homotopy spheres or complex projective spaces. In this paper we define strongly projectively flatness of holomorphic maps into the complex Grassmannian manifold, which is a kind of generalization of holomorphic maps into the complex projective space and prove a rigidity of equivariant strongly projectively flat maps of compact simply connected homogeneous Kähler ma…
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
Classifies 4-manifolds up to connected sum with complex projective planes.
problem Classifying 4-manifolds up to connected sum with complex projective planes.
method Classification based on fundamental group, orientation character, and extension class involving second homotopy group.
result Closed, connected 4-manifolds up to connected sum with copies of the complex projective plane are classified in terms of fundamental group, orientation character, and an extension class involving the second homotopy group.
Proves the Hodge conjecture for complex projective manifolds.
problem Proving the Hodge conjecture for complex projective manifolds.
method Utilizing the Dirac-Dolbeault operator and Nash-Moser generalized inverse function theorem.
result Existence of complex submanifolds whose fundamental classes span rational Hodge classes.
The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the proj…
Research explores flat subspaces in complex projective manifolds using Okounkov bodies.
problem Existence of flat subspaces in complex projective manifolds.
method Utilizes the generalised Legendre transform to the Okounkov body and a result by Schwer--Lytchak.
result Sufficient conditions for the existence of flat subspaces are identified.
Research confirms a conjecture about complex manifolds with total Betti number three.
problem Understanding the minimal total Betti number of closed almost complex manifolds.
method Analyzing properties of almost complex manifolds and using topological results.
result The only simply connected closed complex manifold with total Betti number three is the complex projective plane.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Proves projectivity and ampleness of a Kähler manifold using complex Monge-Ampère equation.
problem Projectivity and ampleness of a compact Kähler manifold with negative Ricci curvature.
method An alternate proof using a C2-estimate for a complex Monge-Ampère equation. result Proves projectivity and ampleness of the Kähler manifold.
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by Ck,α stability proofs. result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
The paper shows that certain bundles have unique volumes.
problem Volume rigidity of specific circle bundles.
method Proving volume rigidity for principal circle bundles over complex projective spaces.
result Principal circle bundles are volume rigid among K-contact manifolds. We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that…
We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kaehler manifold gives rise to a c-projective structure and this is one…
We show that for any complete connected Kähler manifold the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini-Study metric. This est…
Study complex quaternionic manifolds and their c-projective structures.
problem Characterize connections on quaternionic manifolds with specific holonomy.
method Quaternionic Feix--Kaledin construction and analysis of c-projective classes.
result Characterize distinguished connections in quaternionic manifolds.
Classify projective subvarieties in Bogomolov-Guan manifolds using quasi-diagonals.
problem Classify projective subvarieties in non-Kahler holomorphically symplectic manifolds.
method Use quasi-diagonals to classify projective subvarieties.
result Prove that any projective subvariety belongs to a fiber of the Lagrangian fibration.
Classifies holomorphic parabolic geometries on complex manifolds.
problem Classifying holomorphic parabolic geometries on complex manifolds.
method Bounding numerical dimension and using geometric invariants.
result Uncovering foliations and fibrations on smooth projective varieties.
In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…
Smooth lens spaces embed in complex projective planes but not in finite copies.
problem Embedding lens spaces in complex projective planes.
method Analyzes smooth and locally flat embeddings of lens spaces in complex projective planes.
result No finite number of copies of complex projective plane can embed every lens space smoothly.
Lecture notes on curves in complex projective plane from a topological viewpoint.
problem Understanding curves in complex projective plane from a topological perspective.
method Topological analysis of curves in complex projective plane.
result Curves in complex projective plane have unique topological properties.
Extremal Kähler submanifolds of complex projective spaces have natural extensions.
problem Understanding the structure and properties of extremal Kähler submanifolds.
method Analyzing the natural extensions and holomorphic isometric immersions of extremal Kähler submanifolds.
result Every connected extremal Kähler submanifold has a natural extension which is a complete Kähler manifold.
Researchers compute c-projective symmetry algebras for Kähler surfaces.
problem Understanding symmetries in Kähler surfaces.
method Defined and analyzed c-projective vector fields and computed their symmetries.
result Computed c-projective symmetry algebras for Kähler surfaces with essential c-projective vector fields.
Complex and quaternionic projective spaces lack local orthogonal coordinates.
problem Lack of local orthogonal coordinates in complex and quaternionic projective spaces.
method Analysis of Riemannian manifolds and canonical metrics.
result Complex and quaternionic projective spaces do not have local systems of orthogonal coordinates.
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
Constructs negatively curved complete intersections in complex manifolds.
problem Creating compact negatively curved complete intersections.
method Using Donaldson-Auroux theory to construct and prove existence.
result Existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature.
The abstract discusses conditions for four-manifolds to be topologically spheres or complex projective planes.
problem Conditions for four-dimensional compact manifolds to be topologically spheres or complex projective planes.
method Using half-conformally flat Riemannian manifolds and biorthogonal curvature.
result Four-dimensional compact oriented half-conformally flat Riemannian manifolds with sectional curvatures in a specific interval are topologically either S4 or CP2. The paper finds transformation formulas for quaternionic complex structures.
problem Quaternionic projective invariance of k-Cauchy-Fueter complex. method Explicit transformation formulae under mSL(n+1,H). result Quaternionic projectively invariant operator and defining density.
Holomorphic branched Cartan geometry defined on complex manifolds.
problem Defining holomorphic branched Cartan geometry on complex manifolds.
method Using Atiyah bundle and foliations, defining transversely flat branched complex projective geometry.
result Holomorphic branched Cartan geometries on compact manifolds are flat.
Study of complex projective manifolds using arithmetic lattices.
problem Holomorphic convexity for toroidal compactifications of ball quotients.
method Show that Albanese mapping on an étale covering space generates jets on the interior.
result Shafarevich conjecture on holomorphic convexity satisfied in dimension 2 for arithmetic lattices.
It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions.
Study on Gauduchon manifolds finds metrics for projectively flat bundles.
problem Existence of Hermitian-Poisson metrics on projectively flat bundles.
method Heat flow techniques and continuity methods.
result Established a correspondence between Hermitian-Poisson metrics and semi-simplicity.
We develop some theory of double fibration transforms where the cycle space is a smooth manifold and apply it to complex projective space.
Zoll manifolds with entire Grauert tubes are proven to be standard complex projective spaces.
problem Characterizing Zoll manifolds with entire Grauert tubes.
method Isometric comparison to CPn with the canonical metric. result Zoll manifolds of type CPn with entire Grauert tubes are isometric to CPn. The study shows conditions for Kähler manifolds to have rational cohomology of complex projective space.
problem Conditions for Kähler manifolds to have rational cohomology of complex projective space.
method Analyzing the Calabi curvature operator and its positivity conditions.
result Compact Kähler manifolds with specific curvature conditions have rational cohomology of complex projective space.
New proof shows compact Kähler manifolds with positive curvature are complex projective spaces.
problem Characterizing compact Kähler manifolds with positive orthogonal bisectional curvature.
method Siu-Yau's method
result Any n-dimensional compact Kähler manifold with positive orthogonal bisectional curvature is biholomorphic to Pn. The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex vector bundles over an almost-complex manifold is also discussed, as well as appli…
Computes cohomologies of blow-ups and projective bundles.
problem Cohomologies of blow-ups and projective bundles.
method Computes double complex of differential forms on projective bundles and blow-ups.
result Formulas for all cohomologies associated with the complex.
Bounding geodesic length variation for surface projective structures.
problem Understanding how geodesic lengths change under projective structure variations.
method Bounding the derivative of complex length in terms of the Schwarzian norm.
result Application to cone-manifold deformations of hyperbolic 3-manifolds.
Two Kähler metrics on a complex manifold are called c-projectively equivalent if their J-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kähler metrics of arb…
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…
Study of quasi-Kähler metrics on complex manifolds linked to c-projective metrizability.
problem Characterizing quasi-Kähler metrics on almost complex manifolds.
method Analyzing the c-projectively invariant metrizability equation and its solutions.
result New geometries induced by non-degenerate solutions with non-vanishing scalar curvature.