arXiv research

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

Introduces gauge theory for string algebroids, solving Calabi system.

In the present notes we explain the relationship between Calabi-Yau integrable systems and Hitchin systems based on work by Diaconescu-Donagi-Pantev and the author. Besides a review of these integrable systems, we highlight related topics, for example variations of Hodge structures, cameral curves and Slodowy slices, a…

The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.

New metrics solve complex equations on special 3D shapes.

In this paper we construct explicit smooth solutions to the Strominger system on generalized Calabi-Gray manifolds, which are compact non-Kähler Calabi-Yau 3-folds with infinitely many distinct topological types and sets of Hodge numbers.

We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3) holonomy then the eq…

We propose a new construction of compact non-Kähler Calabi-Yau manifolds with balanced metrics and study the Strominger system on them. In particular, we obtain explicit solutions to the Strominger system with degeneracies on $Σ_g\times T^4$, where $Σ_g$ is an immersed minimal surface of genus $g\geq3$ in $T^3$.

Solutions to Strominger system found for square of Kähler class.

Survey on Strominger system and Ricci flow in non-Kähler geometry.

New obstruction found for Hull-Strominger system solutions.

In arXiv:1008.1018 it is shown that a given stable vector bundle $V$ on a Calabi-Yau threefold $X$ which satisfies $c_2(X)=c_2(V)$ can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explic…

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.

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.

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

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

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 …

We study SYZ mirror symmetry in the context of non-Kaehler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric $SU(3)$ systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and…

We completely explore the system of ODE's which is equivalent to the existence of a parallel $Spin(7)$-structure on the cone over a 7-dimensional 3-Sasakian manifold. The one-dimensional family of solutions of this system is constructed. The solutions of this family correspond to metrics with holonomy SU(4) which gener…

Compactifies Calabi-Yau to weak Fano manifolds.

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

The paper studies the twisted Calabi flow on Kähler manifolds.

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

Uniform estimates for Calabi-Yau degenerations proved.

Classifies Calabi hypersurfaces with parallel Fubini-Pick form.

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow pro…

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

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…

The abstract discusses how generalized Calabi-Gray manifolds help solve non-Kähler geometry questions.

Polyhomogeneous expansions for Calabi-Yau metrics near singularities.

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

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…

Holomorphic connections on Calabi-Yau manifolds are flat.

Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.

Sharp diameter bounds for Calabi-Yau degenerations proved.

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

New maximal surfaces solve Bernstein problems.

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.

In this paper, we obtain several a-priori estimates for the Calabi flow on projective bundles admitting the generalized Calabi constructions.

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homog…

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural p…

Study on Calabi-Yau metrics and their degenerations.

This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.