Study rational sections of Picard scheme, leading to Hitchin fibration group description.
problem Understanding rational sections of Picard schemes on smooth projective varieties.
method Analyzing rational sections of relative Picard scheme, proving them from line bundles.
result All rational sections of relative Picard scheme come from line bundles under certain conditions.
Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.
problem Embedding rational ruled surfaces into symplectic manifolds.
method Analyzes symplectic hyperplane sections of rational ruled surfaces.
result Obtains Stein fillability results for rational ruled surfaces.
The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.
problem Characterizing compact Kähler manifolds with nonnegative holomorphic sectional curvature.
method Holonomy principle and geometric properties.
result Compact Kähler manifolds with nonnegative holomorphic sectional curvature are projective and rationally connected.
Positive holomorphic sectional curvature on rational surfaces is characterized by Kähler metrics.
problem Characterizing rational surfaces by the existence of a Kähler metric with positive holomorphic sectional curvature.
method Constructing Kähler metrics on projective manifolds obtained from toric manifolds.
result Every projective manifold obtained from a projective toric manifold by a finite sequence of blow-ups at points admits a Kähler metric with positive holomorphic sectional curvature.
Researchers prove a complex geometric conjecture about certain manifolds.
problem Compact simply connected Riemannian manifolds with nonnegative sectional curvature.
method Assumption of entire Grauert tube and real analytic structure.
result Compact simply connected Riemannian manifolds with entire Grauert tube are rationally elliptic.
The paper shows how uniform RC-positivity on manifolds implies projectivity and rational connectedness.
problem Understanding the conditions for projectivity and rational connectedness in Kähler manifolds.
method Analyzing the properties of uniformly RC-positive metrics and their relationship to projectivity and rational connectedness.
result Uniformly RC-positive metrics on rationally connected manifolds imply projectivity and rational connectedness.
Study on Kodaira fibrations with nontrivial cohomology, proving properties of their structure.
problem Characterizing Kodaira fibrations with specific cohomology properties.
method Analyzing invariant rational cohomology and properties of holomorphic sections.
result Kodaira fibrations with invariant cohomology admit specific coverings and monodromies.
We show that there is a complex structure on the symplectic 4-manifold W4,k obtained from the elliptic surface E(4) by rationally blowing down k sections for 2≤k≤9. And we interpret it via Q-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf.
The paper studies morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature.
problem Establishing structure theorems for compact Kähler manifolds with semi-positive holomorphic sectional curvature.
method Analyzing morphisms to compact Kähler manifolds with pseudo-effective canonical bundles, proving smoothness and isomorphism of fibers, and applying maximal rationally connected fibrations.
result Compact Kähler manifolds with semi-positive holomorphic sectional curvature are rationally connected and have uniformization properties.
The paper proves properties of image structures of fibrations with semi-positive curvature.
problem Structures and images of maximal rationally connected fibrations of projective manifolds with semi-positive holomorphic sectional curvature.
method Generalizes Yang's solution using RC positivity for Yau's conjecture.
result The canonical bundle of images of such fibrations is not big.
Proves section conjecture for curves and surface bundles over various fields.
problem Proving Grothendieck's section conjecture for curves and surface bundles.
method Formulated and proved the section conjecture for stable graphs, used Galois cohomology classes to obstruct sections.
result Proved section conjecture for curves and surface bundles over p-adic and number fields.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…
The paper solves Yau's conjecture and generalizes structure theorems for projective manifolds with semi-positive holomorphic sectional curvature.
problem Projective manifolds with semi-positive holomorphic sectional curvature.
method Combining previous work with the theory of holomorphic foliations, the authors prove that the universal cover of such manifolds is biholomorphic and isometric to a product of a flat metric and a Kähler metric.
result The structure theorem for projective manifolds with semi-positive holomorphic sectional curvature, including the solution for Yau's conjecture.
The paper constructs a hyperbolic 4-manifold with rational homology sphere cusp sections.
problem Constructing a hyperbolic 4-manifold with rational homology sphere cusp sections.
method Constructing a hyperbolic 4-manifold with specified properties.
result The Laplacian on 2-forms on the constructed manifold has purely discrete spectrum.
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …
Study shows no hyperkähler fourfolds in specified conditions.
problem Identifying hyperkähler fourfolds in specific geometric settings.
method Classification and computation of Hodge numbers for fourfolds over rational homogeneous varieties.
result No hyperkähler fourfolds found in the specified conditions.
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the direct sum of the higher homotopy groups of M, tensored with the field of rationa…
Study almost complex structures on six-manifolds using twistor spaces.
problem Understanding the space of almost complex structures on six-dimensional manifolds.
method Using twistor spaces and rational homotopy theory, compute the space of almost complex structures and their homological properties.
result Computed the rational homotopy theoretic minimal model of components of almost complex structures satisfying a Chern number condition.
The formal principle holds for certain globally generated vector bundles on Fano manifolds and smooth rational curves.
problem Proving the formal principle for globally generated vector bundles on compact complex manifolds.
method Applying Cartan's equivalence method to a differential system on the universal family of the Douady space.
result The formal principle is true for Fano manifolds and smooth rational curves under specific conditions.
In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank ≥2 resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic …
We study the conditions under which a Kählerian structure (G,J) of general natural lift type on the cotangent bundle T∗M of a Riemannian manifold (M,g) has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for (T∗M,G,J) to be a Kählerian manifold, is expres…
Abstract: Rational decomposition of homeomorphism spaces for manifolds.
problem Constructing rational homotopy pullback decompositions for homeomorphism spaces.
method Rational homotopy pullback decomposition, nullhomotopy of stabilisation maps, tensor products of truncated operads.
result Rational section of the stabilisation map for homeomorphisms of R^d.
The paper introduces RC-positivity for vector bundles and proves manifold properties.
problem Understanding rational connectedness and positivity in complex manifolds.
method Introducing RC-positivity and proving properties of vector bundles and manifolds.
result Compact Kähler manifolds with positive holomorphic sectional curvature are projective and rationally connected.
Study on metrics of non-negative curvature on vector bundles over specific manifolds.
problem Existence of metrics with non-negative sectional curvature on vector bundles.
method Equivariant structures identified via comparison of equivariant and non-equivariant K-theory, transcribed to rational cohomology, and analyzed using rational homotopy theory.
result Explicit constructions of metrics with non-negative sectional curvature on vector bundles.
We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimensi…
The paper connects Apollonian packings to knot theory and improves link representations.
problem Realizing algebraic links in Apollonian packings.
method Introducing new representations of links in tangency graphs of sphere packings, proving link realizability, and improving upper bounds.
result Any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing.
Simply-connected manifolds of positive sectional curvature M are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipt…
Yang proves manifolds with positive curvature are projective and rationally connected.
problem Proving compact manifolds with positive curvature properties.
method Analyzing hodge numbers and curvature properties.
result Compact Hermitian manifolds with positive curvature are projective and rationally connected.
The article proves estimates on homotopy and cohomology dimensions in fibrations.
problem Estimates on dimensions of homotopy and cohomology groups in fibrations.
method Proves estimates in formal elliptic spaces and specific cases.
result Proves estimates on dimensions of homotopy and cohomology groups in fibrations.
Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification o…
Proves existence of infinite order elements in curvature spaces.
problem Existence of elements in curvature spaces.
method Analyzes homotopy groups of spaces with positive curvature.
result Proves existence of infinite order elements in high-dimensional spaces with positive curvature.
This paper studies torsion obstructions to complex sections on manifolds.
problem Torsion obstructions to finding complex sections on almost complex manifolds.
method Calculations using the Adams-Novikov spectral sequence for Thom spectra.
result Torsion obstructions for finding r complex sections of order p vanish for r<p2−p. The study introduces a new concept of almost nonpositivity for holomorphic sectional curvature and proves its implications on nefness and Miyaoka-Yau inequalities.
problem Understanding the behavior of holomorphic sectional curvature on compact Kähler manifolds.
method Introducing a new notion of almost nonpositivity for holomorphic sectional curvature and proving its implications.
result Compact Kähler manifolds with almost nonpositive holomorphic sectional curvature have nef canonical line bundles, contain no rational curves, and satisfy Miyaoka-Yau type inequalities.
Study describes Abel-Jacobi map for elliptic surfaces, refining cubic-line arrangements topology.
problem Understanding the Abel-Jacobi map on elliptic surfaces and its impact on cubic-line arrangements topology.
method Analyzes the Abel-Jacobi map and its associated rational points on elliptic surfaces, applying it to refine results on cubic-line arrangements.
result Refines the understanding of cubic-line arrangements topology using the Abel-Jacobi map.
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…
In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we …
Paper connects fibre bundles to curvature and homotopy theory.
problem Understanding the relation between curvature and fibre bundles.
method Uses rational homotopy theory to explore fibre bundles.
result Establishes a connection between curvature properties and homotopy invariants.
The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.
problem Understanding the fundamental groups of compact Kähler manifolds with specific curvature properties.
method Analyzing foliations and topological properties to prove a locally trivial fibration structure.
result Compact Kähler manifolds with semi-positive holomorphic sectional curvature admit a locally trivial fibration structure.
We prove that any rational linear combination of Pontryagin numbers that is not a multiple of the signature is unbounded on connected closed oriented manifolds of nonnegative sectional curvature. Combining our result with Gromov's finiteness result for the signature yields a new characterization of the L-genus.
Smooth HP^2 bundle over S^4 with nontrivial A-genus found.
problem Existence of Riemannian metrics with positive sectional curvature.
method Explained existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
result Existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
Study shows unbounded Pontryagin numbers on curved manifolds.
problem Understanding unbounded Pontryagin numbers on curved manifolds.
method Analyzing rational linear combinations of Pontryagin numbers and their relation to the universal elliptic genus.
result Proves existence of unbounded Pontryagin numbers on nonnegatively curved spin manifolds.
Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.
problem Characterizing Kähler-Ricci solitons that can be immersed into complex space forms.
method Proving that a Kähler-Ricci soliton's metric is Einstein if it can be immersed into a complex space form.
result The Kähler-Ricci soliton's metric is an Einstein metric if it can be immersed into a complex space form.
The paper proves conditions for projectivity and rational connectedness of complex manifolds with quasi-positive mixed curvature.
problem Conditions for projectivity and rational connectedness of complex manifolds with quasi-positive mixed curvature.
method Convex combination of Ricci curvature and holomorphic sectional curvature, proving projectivity and rational connectedness under specific curvature conditions.
result Compact complex manifolds with quasi-positive mixed curvature are projective and rationally connected under certain conditions.
Two remarks on curvature properties of Kähler manifolds.
problem Curvature properties of Kähler manifolds.
method Analyzing semi-positive holomorphic sectional curvature and quasi-negative k-Ricci curvature. result For semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration equals the number of non-truly-flat directions. For quasi-negative k-Ricci curvature, the canonical bundle is ample. We show that the moduli space M of holomorphic vector bundles on CP3 that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on R4, called the space of twistor sections. We then use this c…
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM3 manifolds of non-negative curvature. Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.