Projective manifolds with specific bundles are isomorphic to simpler spaces.
problem Characterizing projective manifolds with tangent bundles containing strictly nef subsheaves.
method Analyzing the structure of the tangent bundle and using properties of strictly nef subsheaves.
result Projective manifolds with the described bundles are isomorphic to projective bundles over hyperbolic manifolds or projective spaces.
The paper calculates delta invariants for specific geometric structures.
problem Computing delta invariants for projective bundles and cones of Fano type.
method Provides a precise formula for delta invariants.
result A formula to compute delta invariants for projective bundles and cones of Fano type.
Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.
problem Holomorphic projective structures and bundles on surfaces.
method Generalization of principal bundle of projective 2-frames to branched projective structures.
result Affine spaces of branched projective structures with given branching classes.
Computes the decomposition of rank-three bundles over the projective line with three marked points.
problem Decomposing rank-three bundles over the projective line with three marked points.
method Using the monodromy derivative to compute the roots of the bundles.
result Computes the exact decomposition of rank-three bundles for m=3. 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.
New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
The paper studies conformal and projective structures using 2-frame bundles.
problem Understanding connections between conformal and projective structures.
method Using the dressing field method to obtain local, gauge invariant connections.
result A projective tractor bundle can be defined using the same construction as for conformal structures.
A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From thi…
Researchers create a new metric on complex projective space bundles.
problem Constructing a hyperkähler metric on complex projective space bundles.
method Explicit construction in local coordinates, using holomorphic isomorphism to coadjoint orbits.
result A hyperkähler metric on twisted cotangent bundles of CPn. Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
Study compact Kähler manifolds with pseudo-effective tangent bundles.
problem Characterize compact Kähler manifolds with strongly pseudo-effective tangent bundles.
method New proofs and characterizations of properties of vector bundles.
result Only projective spaces have big tangent bundles.
The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.
problem Conditions for weak symplectic forms on projective limits of Banach bundles.
method Analyzing projective sequences of Banach bundles and applying Darboux Theorem.
result Necessary and sufficient conditions for the Darboux Theorem on projective limits of Banach manifolds.
Paper proves structure of compact Kähler 3-folds with specific bundles.
problem Characterizing compact Kähler 3-folds with nef anti-canonical bundles.
method Minimal Model Program, positivity of direct image sheaves, Q-conic bundles, orbifold vector bundles.
result Compact Kähler 3-folds with nef anti-canonical bundles are essentially one of three types.
The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between line and vector bundles over real projective space.
method F-method for classification and construction of intertwining differential operators.
result Generalizes a classical result of Bol for SL(2,R) and classifies intertwining operators for SL(n,R). 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.
In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety Xn (n>4) has strictly nef Λ2TX, then it is isomorphic to Pn or quadric Qn. We also prove that on elliptic curves, strict…
This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an applic…
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are Z-homology fiber bundles. result Disprove conjecture about homology fiber bundles.
In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle: X admits a smooth fibration X→Y to a flat projective manifold Y such that its general fiber is rationally conn…
Analyzes the moduli space of Higgs bundles to prove its quasi-projectivity.
problem Proving the quasi-projectivity of the moduli space of Higgs bundles.
method Uses analytic methods and the symplectic cut to construct a compactification.
result Proves the quasi-projectivity of the moduli space of Higgs bundles.
We construct a Kaehler structure on the punctured cotangent bundle of the Cayley projective plane whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and we show that the geodesic flow action is holomorphic and is expressed in a quite explicit form. We also give an embedding of the pun…
The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold (X,ω), there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend thi…
In this paper, we obtain several a-priori estimates for the Calabi flow on projective bundles admitting the generalized Calabi constructions.
Explains differences and similarities of strictly nef and ample vector bundles.
problem Characterizing geometry of projective manifolds with strictly nef bundles.
method Brief exposition on strictly nef and ample vector bundles.
result Differences and similarities between strictly nef and ample vector bundles.
Extended Einstein manifolds reveal new symmetries.
problem Understanding symmetries of Einstein manifolds.
method Constructed a line bundle from projective compactification and identified its automorphisms as asymptotic symmetries.
result Asymptotic symmetries identified on extended boundaries of Einstein manifolds.
Geodesic flow mixing on convex projective manifolds proven.
problem Understanding mixing properties of geodesic flow on convex projective manifolds.
method Introduced biproximal unit tangent bundle and proved mixing properties.
result Geodesic flow is topologically mixing on biproximal unit tangent bundle.
Study Miyaoka-Yau inequality for certain projective manifolds.
problem Proving Miyaoka-Yau inequality for specific types of manifolds.
method Using recent work by K.~Zhang and delta-invariant introduced by Fujita and Odaka.
result Established Miyaoka-Yau type inequality for projective manifolds with nef anti-canonical line bundle.
Proves existence of flat connection on theta functions for G-bundles.
problem Existence of flat connections on nonabelian theta functions for G-bundles.
method Proves existence of a flat projective connection on nonabelian theta functions on moduli space of parabolic G-bundles.
result Existence of a flat projective connection on nonabelian theta functions for parabolic G-bundles.
Study Higgs bundles on smooth projective varieties and their restrictions to curves.
problem Interplay between Higgs bundles on smooth projective varieties and their restrictions to curves.
method Investigate the restriction map of Higgs bundles and study branes in moduli spaces.
result Interconnectedness of Higgs bundles and branes on smooth projective varieties and their restrictions.
Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.
problem Characterizing diffeomorphisms between nilmanifolds and smooth quasi-projective varieties.
method Analyzing cohomology and diffeomorphism properties of quasi-projective varieties and nilmanifolds.
result Nilmanifolds are diffeomorphic to trivial bundles over tori under certain conditions.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…
Let H be a semisimple algebraic group. We prove the semistable reduction theorem for μ--semistable principal H--bundles over a {\it smooth projective variety X} defined over the field $\bc$. When X is a {\it smooth projective surface} and H is simple, we construct the algebro--geometric Donaldson--Uhlenbeck…
This research connects Higgs bundles to projective structures via conformal limits.
problem Mapping Higgs bundles to projective structures.
method Using non-abelian Hodge correspondence and conformal limit.
result The family of connections in the conformal limit can be understood as complex projective structures.
In this paper, we propose a method of studying the modified Kahler-Ricci flow on projective bundles and give the explicit equation from the view point of symplectic geometry.
Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle …
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
The paper proves stability of pulled back parabolic bundles on curves.
problem Stability of pulled back parabolic vector bundles on curves.
method Constructing a subbundle and proving stability for a specific case.
result The pullback of a stable parabolic bundle is also stable under certain conditions.
The paper calculates the mapping class group of specific complex projective plane bundles.
problem Computing the mapping class group of complex projective plane bundles.
method Analyzes sphere bundles of real vector bundles over P2. result Calculates the mapping class group for specific examples like Milnor hypersurfaces.
The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.
problem Certifying infinitesimal projective rigidity for hyperbolic once-punctured torus bundles.
method Using twisted Alexander polynomials of representations associated with the holonomy.
result The induced action on the tangent space of the character variety matches the group theoretic action.
It has been argued by Witten and others that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are measured by twisted K-theory. In joint work with Bouwknegt, Carey and Murray it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are or…
Geometrically connects theta functions and WZNW blocks.
problem Identifying connections between theta functions and WZNW blocks.
method Constructing flat connections on bundles of theta functions and WZNW blocks.
result The Tsuchiya-Ueno-Yamada and Knizhnik-Zamolodchikov connections are identified.
Study shows non-existence of almost complex structures on certain sphere bundles over complex projective spaces.
problem Existence of almost complex structures on sphere bundles over complex projective spaces.
method Chern class computations and divisibility properties of characteristic classes.
result Establishes a necessary condition for the non-existence of almost complex structures on certain bundles.
Quillen connection links Riemann surfaces to projective structures.
problem Understanding the relationship between Riemann surfaces and projective structures.
method Using the Quillen connection and Weil-Petersson form on moduli stacks.
result Holomorphic isomorphism between bundles of connections and uniformization.
New examples found of complex manifolds with special metrics.
problem Existence of Kähler-Einstein metrics on certain complex manifolds.
method Using Hultgren's polytope formulation, constructing explicit examples of toric Fano manifolds.
result Found examples of projective bundles that admit coupled Kähler-Einstein metrics but no ordinary Kähler-Einstein metrics.
Generalizes Higgs bundles theory using a vector bundle twist.
problem Extending Higgs bundles theory to incorporate vector bundle twists.
method Defined a Hitchin map and spectral correspondence, stated Hitchin-Kobayashi correspondence.
result Established a theory halfway between curve and higher-dimensional variety Higgs bundles.