arXiv research

Flow analysis leads to metric completion in Kähler geometry.

The Kähler-Ricci flow on certain manifolds collapses to a canonical metric.

The paper proves a version of the SYZ conjecture for hyperkahler manifolds.

Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension

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.

Extends Kollár's result to fibered Calabi-Yau varieties with cohomological assumption.

Two types of nonvanishing results are presented for compact Kähler varieties.

Paper defines new stability and metrics for complex spaces.

Study orders of canonical bundles over graph configuration spaces.

Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.

Paper proves structure of compact Kähler 3-folds with specific bundles.

In 1998, Gompf described a Stein domain structure on the disk cotangent bundle of any closed surface S, by a Legendrian handlebody diagram. We prove that Gompf's Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorp…

We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a compact surface.

Study Miyaoka-Yau inequality for certain projective manifolds.

Uniformizes compact Sasakian manifolds into circle bundles.

The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …

The article describes canonical metrics on holomorphic fibre bundles.

The $\bar{\partial}_{_{J}}$ operator over an almost complex manifold induces canonical connections of type $(0,1)$ over the bundles of $(p,0)$-forms. If the almost complex structure is integrable then the previous connections induce the canonical holomorphic structures of the bundles of $(p,0)$-forms. For $p=1$ we can …

Canonical bundle formula due to Kawamata and others has played fundamental roles in algebraic geometry. We show that the canonical bundle formula has analytic characterization in terms of fiberwise integration, which confirms a folklore conjecture. The proof uses $L^2$ metrics and the valuative equivalence of plurisubh…

Gradient and eigenvalue estimates for Kähler manifolds' canonical bundle.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.

This note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz to general Kähler classes. We give some examples of explicit families, in partic…

For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between …

We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach.

In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi ansatz technique. We use this approach to provide several e…

Study on Kähler manifolds with non-positive mixed curvature and its implications.

We give a representation of canonical vector bundles over Grassmannian manifolds as non-compact affine symmetric spaces as well as their Cartan model in the group of the Euclidean motions.

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…

Geometric equation defines canonical metrics on vector bundle families.

This paper introduces $\infty$- and $n$-fold vector bundles as special functors from the $\infty$- and $n$-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of $n$-fold vector bundles and we prove that any $n$-fold vector bundle admits a non-canonical isomorphism to a decomposed …

The paper explores geometric structures on Weil bundles and their canonical lifts.

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.

In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the c…

Canonical maps connect complex structures to Hitchin components.

We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.

The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.

Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.

We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group $SU(n)$, for all Kähler classes. These metrics are natural generalizations of the metrics of Candelas-de la Ossa on the conifold, Pando Zayas-Tseytlin on the canonical bundle over $\mathbb{CP…

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical thresh…

In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the $L^2$-norm of the scalar curvature along the Kähler-Ricci flow.

The paper proves a numerical condition for solving complex Hessian quotient equations with Calabi symmetry.

New equivalences found between graded supermanifolds and vector bundles.

The paper shows how compact Kähler manifolds with a special bundle can be broken down into simpler parts.

Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.