Related papers: Collapsing hyperk\"ahler manifolds

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat K\"ahler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the…

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when…

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration…

Given a holomorphic Lagrangian fibration of a compact hyperkahler manifold, we use the differential geometry of the special Kahler metric that exists on the base away from the discriminant locus, and show that the pullback of the tangent…

We study Calabi-Yau metrics on a projective manifold in K\"ahler classes converging to a semiample class given by a fibration. We show that the Gromov-Hausdorff limit of the metrics is homeomorphic to the base of the fibration and in…

In this paper, we study the collapsing behaviour of negative K\"{a}hler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space…

In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the…

We investigate the collapsing geometry of hyperk\"ahler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperk\"ahler metrics on the K3 manifold is isometric to one of…

One of the main results of the paper arXiv:1108.0967 by Gross-Tosatti-Zhang establishes estimates on the collapsing of Ricci-flat Kahler metrics on holomorphic torus fibrations. We remove a projectivity assumption from these estimates and…

In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of…

We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity…

We study the collapsing of Calabi-Yau metrics and of Kahler-Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov-Hausdorff limit of the Kahler-Ricci flow when the divisorial part of the discriminant locus…

The aim of this paper is to study pointed Gromov-Hausdorff Convergence of sequences of K\"ahler submanifolds of a fixed K\"ahler ambient space. Our result shows that lower bounds on the scalar curvature imply convergence to a smooth…

We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial K\"ahler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold…

Let $\mathcal{K}(n, V)$ be the set of $n$-dimensional compact Kahler-Einstein manifolds $(X, g)$ satisfying $Ric(g)= - g$ with volume bounded above by $V$. We prove that after passing to a subsequence, any sequence $\{ (X_j,…

For the sake of hyperk{\"a}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{\"a}hler manifolds are real dimension $4n$ non-compact hyperk{\"a}hler manifolds which are quaternion analog…

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahler metric) as one approaches a large complex…

For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat K\"ahler metrics such that curvatures are uniformly bounded away from singular fibers, and which…

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic…

Let $\pi: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact K\"ahler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique K\"ahler-Einstein…