Related papers: What is... a G_2 manifold?

This is a survey paper. We explain the known constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy G2. One method uses resolutions of singularities of appropriately chosen…

We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost K\"ahler manifolds. We give an explicit non-compact example of an Einstein almost cok\"ahler manifold that is not cok\"ahler. We prove that compact…

We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat Kahler metrics, with an emphasis on the case when the metrics are volume collapsing.

Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth…

We introduce the concept of G2(2)-structure on an orientable 3-manifold M using the setting of generalized geometry of type Bn, study their local deformation by making use of a Moser-type argument, and give a description of the cone of…

This paper proposes sG-hyperbolicity as a new tool for studying hyperbolicity on complex manifolds. It demonstrates that this notion leads to a wider class of divisorially hyperbolic manifolds compared to balanced hyperbolicity. We also…

We continue to develop our method for effectively computating the special K\"ahler geometry on the moduli space of Calabi-Yau manifolds. We generalize it to all polynomial deformations of Fermat hypersurfaces.

This paper investigates the generalizations and applications of weakly $p$-K\"ahler hyperbolic manifolds.

We clarify the recently proposed method to compute a Special K\"ahler metric on a Calabi-Yau complex structures moduli space that uses the fact that the moduli space is a subspace of specific Frobenius manifold. We apply this method to…

Using an algebraic orbifold method, we present non-commutative aspects of $G_2$ structure of seven dimensional real manifolds. We first develop and solve the non commutativity parameter constraint equations defining $G_2$ manifold algebras.…

In this paper we investigate the geometry of Calibrated submanifolds and study relations between their moduli-space and geometry of the ambient manifold. In particular for a Calabi-Yau manifold we define Special Lagrangian submanifolds for…

In this paper we introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi-Yau metric. New equation differs from the Monge--Ampere equation considered by Calabi and Yau.

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the…

This is the first nontrivial construction to date of instantons over a compact manifold with holonomy exactly $G_2$. The HYM connections on asymptotically stable bundles over Kovalev's noncompact Calabi-Yau 3-folds, obtained in the first…

The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.

Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form \phi. A choice of 2-plane field \Lambda on M (which always exits) splits the tangent bundle of M as a direct sum of a…

Recently, at least 50 million of novel examples of compact $G_2$ holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry…

This paper has two purposes. First it partially extends the result in the author's previous work concerning the asymptotic expansion of the Tian-Yau metrics, by considering a slightly larger class of quasi-projective manifolds. This text is…

Notions of self-dual and anti self-dual almost quaternionic structures are introduced. The complete classification of self-dual and anti self-dual generalized Kaehler manifolds is obtained.

Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the…