arXiv research

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.)

Study noncommutative deformations of Calabi-Yau threefolds.

We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.

Holomorphic connections on Calabi-Yau manifolds are flat.

To pave the way for the journey from geometry to conformal field theory (CFT), these notes present the background for some basic CFT constructions from Calabi-Yau geometry. Topics include the complex and Kaehler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss CFT const…

This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students i…

Study shows continuity of non-Kähler Calabi-Yau conifold transitions.

We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.

Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.

Weil-Petersson volumes vary continuously with weighted points on a projective line.

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

Constructs special Lagrangian 3-spheres in non-Kähler compact threefolds.

We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact Kähler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projectiv…

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

Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.

We generalize Calabi-Yau 3-folds from the special Lagrangian perspective. More precisely, we study SU(3)-structures which admit as "nice" a local special Lagrangian geometry as the flat $\mathbf{C}^3$ or a Calabi-Yau structure does. The underlying almost complex structure may not be integrable. Such SU(3)-structures ar…

We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the par…

Two Calabi-Yau theorems for Kähler manifold degenerations.

We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the sub…

Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.

Neural networks approximate Calabi-Yau metrics and curvature.

Study collapsing Calabi-Yau metrics and flows on fiber spaces.

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

Constructs hyperbolic affine spheres and 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.

In this paper we study the geometry of manifolds with vector cross product and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and co…

Survey on complex Monge-Ampère equations in Hermitian contexts.

We construct a family of Calabi-Yau metrics on $\C^3$ with properties analogous to the Taub-NUT metric on $\C^2$, and construct a family of Calabi-Yau 3-fold metric models on the positive and negative vertices of SYZ fibrations with properties analogous to the Ooguri-Vafa metric.

New complete Calabi-Yau metrics found in complex space.

New Spin(7) manifolds created from Calabi-Yau bundles.

Study explores unstable 3-forms on Calabi-Yau 3-folds.

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow c…

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…

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

Solves generalized Kähler Calabi-Yau problem on compact manifolds.

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…

Survey on metric SYZ conjecture and non-archimedean geometry.

We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some …

New octonionic Kähler metrics solve an octonionic Calabi-Yau theorem.

We extend the notion of (branched) holomorphic Cartan geometry on a complex manifold to the context of Sasakian manifolds. Branched holomorphic Cartan geometries on Sasakian Calabi-Yau manifolds are investigated.

Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.

In recent work N. Hitchin introduced the concept of "generalised geometry". The key feature of generalised structures is that that they can be acted on by both diffeomorphisms and 2-forms, the so-called $B$-fields. In this lecture, we give a basic introduction and explain some of the fundamental ideas. Further, we disc…

We find sufficient conditions for a principal toric bundle over compact Kähler manifolds to admit Calabi-Yau connections with torsion. With the aids of a topological classification, we construct such geometry on $n(S^2\times S^4)#(n+1)(S^3\times S^3)$

Machine learning predicts topological properties of Calabi-Yau manifolds.

Stability of Type IIA flow ensures Kähler properties of Calabi-Yau 3-folds.

Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve counting on a Calabi--Yau manifold is related to a conjectured invariant, only depending on the complex str…

Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.

In this paper, we study the Chern classes on the moduli space of polarized Calabi-Yau manifolds. We prove that the integrations of the invariants of the curvature of the Weil-Petersson metric are finite. In some special cases, they are even rational numbers.