arXiv research

New model for Calabi-Yau-$\mathbb{X}$ categories using decorated marked surfaces.

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…

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

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.

Complete Calabi-Yau metrics made on special 3D spaces.

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

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

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-Kähler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space $M…

Computes link invariants in real projective 3-space using topological vertex.

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

New one-parameter families of $SU(2)^2$-invariant instantons found on Calabi-Yau 3-folds.

The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.

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

Method finds approximate Ricci-flat metrics on Calabi-Yau manifolds.

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

Constructs hyperbolic affine spheres and Calabi-Yau metrics.

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

The study constructs balanced and rigid curves on specific types of hypersurfaces and complete intersections.

Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.

Researchers found a Calabi-Yau structure and constructed a Bargmann type transformation on the Cayley projective plane.

The paper proves a Calabi-Yau structure on complexifications of rank two symmetric spaces.

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…

Machine learning predicts topological properties of Calabi-Yau manifolds.

Let S be a K3 surface that admits a non-symplectic automorphism $ρ$ of order 3. We divide $S\times \mathbb{P}^1$ by $ρ\timesψ$ where $ψ$ is an automorphism of order 3 of $\mathbb{P}^1$. There exists a threefold ramified cover of a partial crepant resolution of the quotient that is a Calabi-Yau orbifold. We compute the …

Study on non-Kähler Calabi-Yau geometries on 3-folds with constraints.

This paper classifies Calabi-Yau manifolds near cones.

Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.

We produce special Lagrangian $T^n$-fibrations on the generic regions of some Calabi-Yau hypersurfaces in the Fermat family $X_s=\{ Z_0\ldots Z_{n+1}+ e^{-s} ( Z_0^{n+2}+ \ldots Z_{n+1}^{n+2} ) =0 \}\subset \mathbb{CP}^{n+1}$ near the large complex structure limit $s\to +\infty$.

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

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also sho…

The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.

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

Calculates Laplacian spectra on Calabi-Yau hypersurfaces.

Let $({\bf X},ω_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf X}=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological space. We prove that …

New explicit Calabi-Yau metrics and Kähler-Ricci solitons found on complex n-space.

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

The paper proves an isomorphism between $tt^*$ structures of Landau-Ginzburg and Calabi-Yau models.

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…

Machine learning approximates Calabi-Yau Hodge numbers from weight systems.

In this paper we define the analogue of Calabi--Yau geometry for generic $D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in $E_{7(7)}\times\mathbb{R}^+$ gene…

New cohomology $η$ for deformed Sasaki-Einstein manifolds derived from Dolbeault cohomology.

Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.

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…

Study on holonomy of Obata connection on Joyce hypercomplex manifolds.

Constructs complete metrics and solitons on complex vector bundles.

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

We construct solutions to the Strominger system on a class of noncompact Calabi-Yau 3-folds. These spaces include $\mathbb{C}^3$ and resolved conifold $\mathcal{O}(-1,-1)$ as special examples.

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $\varphi$ induced by the Calabi-Yau $3$-orbifold basic geometry. We disti…