arXiv research

The paper classifies certain singular projective varieties with specific properties.

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

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

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

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

Constructs unique bases for CY varieties over valued fields.

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…

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 …

Constructs projective moduli spaces for Calabi-Yau pairs.

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

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

Develops moduli theory for Calabi-Yau pairs, constructing a projective space.

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.

We construct BCOV invariant for Calabi-Yau pairs. The construction covers the classical BCOV invariant and certain equivariant BCOV invariant. The BCOV invariant obtained is expected to be well-behaved under birational equivalence.

Paper proves uniqueness of special Lagrangian pair in Calabi-Yau 3-fold.

We give a differential-geometric construction and examples of Calabi-Yau threefolds, at least one of which is {\it{new}}. Ingredients in our construction are {\it admissible pairs}, which were dealt with by Kovalev in \cite{K03} and further studied by Kovalev and Lee in \cite{KL11}. An admissible pair $(\overline{X},D)…

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 polystability theory connects Calabi-Yau varieties to gravitational instantons.

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…

We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles…

Extends BCOV invariant to pairs of Calabi-Yau manifolds and pluricanonical divisors.

We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first d…

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.

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…

Given a complex 4-fold $X$ with an (Calabi-Yau 3-fold) anti-canonical divisor $Y$, we study relative Donaldson-Thomas invariants for this pair, which are elements in the Donaldson-Thomas cohomologies of $Y$. We also discuss gluing formulas which relate relative invariants and $DT_{4}$ invariants for Calabi-Yau 4-folds.

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…

Perfect pairing for tropical cycles on integral affine manifolds.

Construct algorithms for Frobenius manifolds and residue pairings on Calabi-Yau varieties.

The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.

Proves SYZ mirror symmetry for del Pezzo and rational elliptic surfaces.

The paper classifies and studies symplectic and contact properties of circular spherical divisors.

Previously the two of the authors defined a notion of dual Calabi-Yau manifolds in a G_2 manifold, and described a process to obtain them. Here we apply this process to a compact G_2 manifold, constructed by Joyce, and as a result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known to be mirror dual…

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…

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

Alternative proof of semipositivity and nefness for K-semistable log-Fano pairs.

The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.

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

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…

Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.

Analyzes structure of log smooth pairs when equality holds in Bogomolov-Gieseker inequality.

We study several questions involving relative Ricci-flat Kähler metrics for families of log Calabi-Yau manifolds. Our main result states that if $p:(X,B)\to Y$ is a Kähler fiber space such that $\displaystyle (X_y, B|_{X_y})$ is generically klt, $K_{X/Y}+B$ is relatively trivial and $p_*(m(K_{X/Y}+B))$ is Hermitian fla…

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is called the BCOV invariant. In this paper, we consider a pair $(X,Y)$, where $X$ is a compact Kaehler manifold and $Y\in\big|K_X^m\big|$ with $m\in\mathbb{Z}\backslash\{0,-1\}$. We extend the BCOV invariant to suc…

The paper studies orbifold splice quotients and log covers of surface pairs.

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 …

The curvature properties of a specific type of 6-manifold are explored.

Study shows polystability of tangent and canonical sheaves on Kähler-Einstein log Fano pairs.