arXiv research

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…

Proves certain Calabi-Yau varieties are projective.

Introduces new limit spaces for degenerating Calabi-Yau families.

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.

The paper classifies certain singular projective varieties with specific properties.

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

We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds …

Study of line bundles on spherical varieties leads to Calabi-Yau metrics.

The study shows boundedness of certain fibered varieties in algebraic geometry.

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

Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.

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

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

In this paper we construct monodromy representing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.

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.

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…

In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.

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

In this note, we obtain existence results for complete Ricci-flat Kahler metrics on crepant resolutions of singularities of Calabi-Yau varieties. Furthermore, for certain asymptotically flat Calabi-Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial…

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

Researchers prove birational invariance of BCOV invariant using motivic integration.

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

In this paper, we construct a completion of the moduli space for polarized Calabi-Yau manifolds by using Ricci-flat Kähler-Einstein metrics and the Gromov-Hausdorff topology, which parameterizes certain Calabi-Yau varieties. We then study the algebro-geometric perperties and the Weil-Petersson geometry of such completi…

Constructs unique bases for CY varieties over valued fields.

Complete Calabi-Yau metrics on abelian fibrations over complex space.

We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric variety near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi-Yau hypersurface. The construction is based on the deformati…

Stability results for complex Monge-Ampère equations in various classes.

Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.

This paper is a sequel to arXiv:1012.2940. We further investigate the Gromov-Hausdorff convergence of Ricci-flat Kähler metrics under degenerations of Calabi-Yau manifolds. We extend Theorem 1.1 in arXiv:1012.2940 by removing the condition on existence of crepant resolutions for Calabi-Yau varieties.

Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.

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 non-Kähler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves.

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

Survey on complex Monge-Ampère equations in Hermitian contexts.

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…

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

In this article we investigate diffeomorphism classes of Calabi-Yau threefolds. In particular, we focus on those embedded in toric Fano manifolds. Along the way, we give various examples and conclude with a curious remark regarding mirror symmetry.

Researchers extend period maps for Calabi-Yau types using modified Kato-Nakayama-Usui construction.

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…

We survey recent developments in the study of SYZ mirror symmetry for compact toric and toric Calabi-Yau varieties, with a special emphasis on works of the author and his collaborators.

Non-Archimedean balanced metrics approximate cscK metrics for totally degenerate abelian varieties

Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory…

Complex projective manifolds without rational curves are quotients of Abelian varieties.

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…

New metrics found on non-Kähler Calabi-Yau manifolds.

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C$ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of nonc…

The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.