arXiv research

Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.

Study geometric properties of log Calabi-Yau manifolds, focusing on Fano manifolds with smooth or two proportional components.

Log minimality proven for weak K-moduli compactifications of Calabi-Yau varieties.

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

Study shows how volume forms on degenerating varieties converge to a non-Archimedean measure.

New polystability theory connects Calabi-Yau varieties to gravitational instantons.

Study solves Monge-Ampère equation for complete Calabi-Yau metrics.

The paper classifies certain singular projective varieties with specific properties.

Solves non-Archimedean Calabi-Yau equation on complex log pairs.

Researchers prove birational invariance of BCOV invariant using motivic integration.

In this paper, by applying Greene-Shapere-Vafa-Yau semi-flat metric, we give a new proof of closed formula of Weil-Petersson metric on moduli space of Calabi-Yau varieties.

This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certai…

Study on Kähler-Einstein metrics on log Calabi-Yau families, proving local triviality and discrediting a metric conjecture.

The study finds infinitely many special Lagrangian submanifolds in certain complex manifolds.

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…

Constructs unique bases for CY varieties over valued fields.

The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.

We show that every coarse moduli space, parametrizing complex special linear rank two local systems with fixed boundary traces on a surface with nonempty boundary, is log Calabi-Yau in that it has a normal projective compactification with trivial log canonical divisor. We connect this to a novel symmetry of generating …

Sufficient condition for log-continuity of complex Monge-Ampère solutions.

Holomorphic torsion invariant for log-Enriques surfaces derived from Borcherds products.

Let $S$ be a smooth projective variety and $Δ$ a simple normal crossing $\mathbb{Q}$-divisor with coefficients in $(0,1]$. For any ample $\mathbb{Q}$-line bundle $L$ over $S$, we denote by $\mathscr{E}(L)$ the extension sheaf of the orbifold tangent sheaf $T_S(-\log(Δ))$ by the structure sheaf $\mathcal{O}_S$ with the …

Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety…

Study constructs complete Kahler-Einstein metrics near isolated log-canonical singularities.

New non-Kähler 3-folds constructed via log conifold transitions.

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invarian…

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

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. …

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

Uniform estimates for Calabi-Yau degenerations proved.

Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.

Researchers use gluing method to describe collapsing Calabi-Yau metrics on K3 fibred 3-folds.

Polyhomogeneous expansions for Calabi-Yau metrics near singularities.

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

The main goal of this paper is to prove the polystability of the logarithmic tangent sheaf $\mathscr T_X(-D)$ of a log canonical pair $(X,D)$ whose canonical bundle $K_X+D$ is ample, generalizing in a significant way a theorem of Enoki. We apply this result and the techniques involved in its proof to get a version of t…

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…

In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…

Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.

In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply con…

New Calabi-Yau metrics converge polynomially to Calabi model space.

Holomorphic connections on Calabi-Yau manifolds are flat.

Sharp diameter bounds for Calabi-Yau degenerations proved.

Survey on decorated marked surfaces for Calabi-Yau categories.

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

Constructs BCOV invariant for Calabi-Yau pairs.

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.

Study on Calabi-Yau metrics and their degenerations.