arXiv research

The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.

In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.

Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.

New examples of Calabi-Yau 3-folds with unique properties.

Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.

This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.

New Calabi-Yau metrics found on complex symmetric spaces.

New Calabi-Yau metrics with conical singularities are created near complex lines.

Polyhomogeneous expansions for Calabi-Yau metrics near singularities.

We construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work…

Extends results on smoothability of singular Fano and Calabi-Yau varieties.

Proves certain Calabi-Yau varieties are projective.

We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on…

This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students i…

This paper is a follow-up to an earlier paper math.DG/0410260 on desingularizations of Calabi-Yau 3-folds with a conical singularity. In math.DG/0410260 we study Calabi-Yau 3-folds M_0 with a conical singularity at x modelled on some Calabi-Yau cone V, and construct a desingularization of M_0 by gluing in an Asymptotic…

New Calabi-Yau metrics constructed with detailed geometry at infinity.

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

Study deformations of compact Calabi-Yau conifolds with singularities.

We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general …

We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the sub…

On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics…

Researchers prove birational invariance of BCOV invariant using motivic integration.

The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.

Study of $G_2$-structures with isolated singularities and bounded torsion.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces …

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in t…

Study solutions and singularities of G2-structures flows on specific manifolds.

Study collapsing Calabi-Yau metrics and flows on fiber spaces.

Neural networks approximate Calabi-Yau metrics and curvature.

Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.

For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and Kerr have expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV invariants, s…

Continuity of Kähler-Einstein potentials at singularities proven.

Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canoni…

We construct ALE Calabi-Yau metrics with cone singularities along the exceptional set of resolutions of $\mathbb{C}^n / Γ$ with non-positive discrepancies. In particular, this includes the case of the minimal resolution of two dimensional quotient singularities for any finite subgroup $Γ\subset U(2)$ acting freely on t…

Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.

Motivated by the study of collapsing Calabi-Yau threefolds with a Lefschetz K3 fibration, we construct a complete Calabi-Yau metric on $\mathbb{C}^3$ with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi-Yau metric has singular tangen…

Survey on complex Monge-Ampère equations in Hermitian contexts.

We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.

In this paper, we study the generalized Lagrangian mean curvature flow in almost Einstein manifold proposed by T. Behrndt. We show that the singularity of this flow is characterized by the second fundamental form. We also show that the rescaled flow at a singularity converges to a finite union of Special Lagrangian con…

For each $n\ge 3$, we construct on $\mathbb{C}^n$ examples of complete Calabi-Yau metrics of Euclidean volume growth having a tangent cone at infinity with singular cross-section.

The paper classifies certain singular projective varieties with specific properties.

In this note we give an overview of some applications of the Calabi-Yau theorem to the construction of singular positive (1,1) currents on compact complex manifolds. We show how recent developments allow us to give streamlined proofs of existing results, as well as new ones.

In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized La…

Study shows how neck pinches occur in Lagrangian flows and their continuation.

Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.