Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
Survey on hypothetical complex structure on 6-sphere.
problem Understanding the algebraic dimension and biholomorphisms of a hypothetical complex 6-sphere.
method Discussion of existing results and examples.
result Overview of Peternell--Campana--Demailly's result on algebraic dimension and Huckleberry--Kebekus--Peternell's on biholomorphisms.
Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…
Study bounds on Monge-Ampère volumes for degenerate complex equations.
problem Bounds on volumes of Monge-Ampère measures for degenerate complex equations.
method Fine use of quasi-plurisubharmonic envelopes.
result Established a transcendental version of the Grauert-Riemenschneider conjecture.
Transcendental holomorphic Morse inequalities aim at characterizing the positivity of transcendental cohomology classes of type (1,1). In this paper, we prove a weak version of Demailly's conjecture on transcendental Morse inequalities on compact Kähler manifolds. And as a consequence, we partially improve a result o…
After a historical discussion of classical uniformisation results for Riemann surfaces, of problems appearing in higher dimensions, and of uniformisation results for projective manifolds with trivial or ample canonical bundle, we introduce the basic technical concepts and sketch the ideas of the proofs for recent unifo…
The study of projective varieties with nef anticanonical divisors and log terminal singularities.
problem Understanding the structure and properties of projective varieties with specific divisor conditions.
method Analyzing the Albanese map and MRC fibration for klt projective varieties, showing locally constant fibrations and product decompositions.
result Generalization of results for smooth projective varieties to the klt case, including decomposition into rationally connected and projective varieties with trivial canonical divisor.
In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold (X,ω) with the Hermitian metric ω satisfying the Gauduchon and Astheno-Kähler conditions. We prove that numerically flatness is equivalent to numerically effectiveness with vanishing first Chern number, semistabl…
The article characterizes complex torus quotients with numerical conditions.
problem Characterizing quotients of complex tori by finite groups.
method Numerical vanishing condition on Chern classes, Bogomolov-Gieseker inequality for singular spaces.
result Generalization of previous results in projective and three-dimensional settings.
We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tange…
Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.
problem Characterizing holomorphic automorphisms with high entropy on hyperkähler manifolds.
method Using Jensen's inequality and properties of stable and unstable distributions, the authors show uniform contraction and expansion, leading to the conclusion that the manifold is birational to a torus quotient.
result Holomorphic automorphisms with high entropy on hyperkähler manifolds are Kummer examples.
We prove the qualitative part of Demailly's conjecture on transcendental Morse inequalities for differences of two nef classes satisfying a numerical relative positivity condition on an arbitrary compact Kähler (and even more general) manifold. The result improves on an earlier one by J. Xiao whose constant 4n featur…
The paper proves structures for complex projective varieties with certain tangent bundle properties.
problem Characterizing complex projective varieties based on properties of their tangent bundles.
method Analyzing the positivity of exterior powers of tangent bundles and using étale covers.
result Complex projective varieties with nef exterior powers of tangent bundles are Fano fiber spaces over Abelian varieties.
The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.
problem Analyzing properties of algebraic fibre spaces with strictly nef relative anti-log canonical divisors.
method Using projective klt pairs and fibration techniques, the paper proves locally constant fibration properties and rational connectedness.
result The fibration is locally constant with rationally connected fibers, and the base is a canonically polarized hyperbolic projective manifold.
Paper extends Hodge correspondence to singular Kähler spaces.
problem Establishing Hodge correspondence over Kähler spaces with singularities.
method Using equivalence of polystable Higgs bundles and semi-simple flat bundles over regular loci, and descent theorem for semistable Higgs bundles.
result Non-abelian Hodge correspondence established over compact Kähler klt spaces and their regular loci.
Study shows symplectic hypersurfaces transform complex projective spaces.
problem Transforming symplectic manifolds into complex projective spaces.
method Hamiltonian circle action and invariant hypersurface analysis.
result Symplectic manifolds and hypersurfaces transform into homotopy complex projective spaces.
The paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.
Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.
problem Investigate Higgs bundles and flat connections on compact Sasakian manifolds.
method Introduce quasi-regularity and regularity of vector bundles, relate to orbibundles, extend non-abelian Hodge correspondence.
result Extend non-abelian Hodge correspondence to quasi-regular Sasakian manifolds.