The paper proves a version of the SYZ conjecture for hyperkahler manifolds.
problem Proving the SYZ conjecture for hyperkahler manifolds with specific conditions.
method Introduced a Teichmuller space to parametrize pairs of hyperkahler manifolds up to isotopy, used a version of the global Torelli theorem to deduce the deformation invariance of semiampleness.
result Proved the SYZ conjecture for hyperkahler manifolds with specific conditions.
Flow analysis leads to metric completion in Kähler geometry.
problem Analyzing Kähler-Ricci flows on compact manifolds.
method Normalized Kähler-Ricci flow convergence to Gromov-Hausdorff limits.
result Metric completion of twisted Kähler-Einstein metric.
Extends Kollár's result to fibered Calabi-Yau varieties with cohomological assumption.
problem Deformations of fibered Calabi-Yau varieties under cohomological constraints.
method Hodge theoretic techniques and T1-lifting criterion of Kawamata-Ran. result Small deformations of fibered Calabi-Yau varieties remain fibered.
Paper defines new stability and metrics for complex spaces.
problem Stability and metrics for complex spaces with big cohomology classes.
method Introduces slope stability and Hermitian-Einstein metrics for big cohomology classes.
result Kobayashi Hitchin correspondence and Bogomolov Gieseker inequality proved.
Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.
problem Understanding Calabi-Yau metrics on converging manifolds.
method Analysis of Gromov-Hausdorff limits of metrics on Calabi-Yau fibrations.
result Gromov-Hausdorff limit is homeomorphic to the base of the fibration and discriminant locus has high Hausdorff codimension.
The paper proves a numerical condition for solving complex Hessian quotient equations with Calabi symmetry.
problem Solvability of complex Hessian quotient equations with specific symmetry.
method Proving a numerical condition and proposing a conjecture on existence of k-subharmonic representatives. result Numerical condition ensures solvability of complex Hessian quotient equations.
Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension
problem Convergence of scalar curvature in Kähler-Ricci flow
method Uniform μ-entropy or uniform Sobolev inequality result Scalar curvature converges to negative Kodaira dimension
The Kähler-Ricci flow on certain manifolds collapses to a canonical metric.
problem Understanding the behavior of Kähler-Ricci flow on compact manifolds.
method Asymptotic expansion of evolving metrics and analysis of the Iitaka fibration.
result The flow collapses to a canonical metric on the base of the Iitaka fibration.
We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so…
Study of twisted Kähler-Einstein metrics on Calabi-Yau spaces with singularities.
problem Understanding the collapsed Gromov-Hausdorff limits of Calabi-Yau spaces.
method Analyzing the geometry of twisted Kähler-Einstein metrics on holomorphic fiber spaces.
result Proving the existence of conical-type singularities in the base of fiber spaces.
The paper proves conditions for the Abundance conjecture in minimal projective klt pairs.
problem Proving the Abundance conjecture for minimal klt pairs with non-zero canonical bundle.
method Analyzing asymptotic behavior of multiplier ideals and properties of supercanonical currents.
result Supercanonical currents are central to proving the Abundance conjecture.
Two types of nonvanishing results are presented for compact Kähler varieties.
problem Deriving geometric consequences from numerical information in Kähler geometry.
method Analyzing non-uniruled varieties and hyperkähler manifolds to establish nonvanishing results for adjoint and nef bundles.
result Strong abundance-type results are obtained in dimension 4.