Related papers: HyperK\"ahler Potentials in Cohomogeneity Two

Let $X$ be a toric $\mbox{hyperK\"ahler}$ manifold. The purpose of this note is to describe the topological $K$-ring $K^*(X)$ of $X$. We give a presentation for the topological $K$-ring in terms of generators and relations similar to the…

Let $M$ be a Kaehler manifold, and consider the total space $T^*M$ of the cotangent bundle to $M$. We show that in the formal neighborhood of the zero section $M \subset T^*M$ the space $T^*M$ admits a canonical hyperkaehler structure,…

We develop an efficient procedure for counting holomorphic functions on a hyperKahler cone that has a resolution as a cotangent bundle of a homogeneous space by providing a formula for computing the corresponding Highest Weight Generating…

Let $(M,h)$ be a Hermitian manifold and $\psi$ a smooth weight function on $M$. The $\partial$-complex on weighted Bergman spaces $A^2_{(p,0)}(M,h, e^{-\psi})$ of holomorphic $(p,0)$-forms was recently studied in [[10] and [9]. It was shown…

We revisit construction of the Atiyah-Hitchin manifold in the generalized Legendre transform approach. This is originally studied by Ivanov and Rocek and is subsequently investigated more by Ionas, in the latter of which the explicit forms…

Associated with a smooth, $d$-closed $(1, 1)$-form $\alpha$ of possibly non-rational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0,…

In this paper, we describe an example of a hyperkaehler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach…

In this paper we address the following question regarding the regularity of geodesics in the space of K\"ahler potentials. Given a geodesic which is highly regular, and has smooth boundary value, can we expect that it is actually smooth? We…

The paper is part of an attempt of understanding non-K\"ahler threefolds. We start by looking at compact complex non-K\"ahler threefolds with algebraic dimension two and admitting locally conformally K\"ahler metrics. Under certain…

The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle $T^*\mathbb{CP}(1)$ of the complex projective line $\mathbb{CP}(1) \simeq \mathbb{S}^2$, which extends the Fubini-Study metric of $\mathbb{CP}(1)$. By…

We give a basis for the Hochschild cohomology ring of tame Hecke algebras. We then show that the Hochschild cohomology ring modulo nilpotence is a finitely generated algebra of Krull dimension 2, and describe the support varieties of…

The Beauville-Fujiki relation for a compact Hyperk\"ahler manifold $X$ of dimension $2k$ allows to equip the symmetric power $\text{Sym}^kH^2(X)$ with a symmetric bilinear form induced by the Beauville-Bogomolov form. We study some of its…

In the setting of super forms developed in a previous article by the author, we introduce the notion of $\mathbb{R}$-K\"ahler metrics on $\mathbb{R}^{n}$. We consider existence theorems and $L^{2}-$estimates for the equation…

Let $G$ be an even dimensional, connected, abelian Lie group and $(\mathcal{A}^\infty,G,\alpha,\tau)$ be a $C^*$-dynamical system equipped with a faithful $G$-invariant trace $\tau$. We show that whenever it determines a…

We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)K\"ahler metric. Furthermore we show that the (pseudo-)K\"ahler metrics defined on some domain in…

The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry', related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the…

In \cite{B3}, Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space $M$ of stability conditions of a $CY_3$ triangulated category. Given a non-degeneracy assumption, a feature of this structure…

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahler metric) as one approaches a large complex…

We show how certain topological properties of co-K{\"a}hler manifolds derive from those of the K\"ahler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-K\"ahler…

This paper gives a partial desingularisation construction for hyperk\"ahler quotients and a criterion for the surjectivity of an analogue of the Kirwan map to the cohomology of hyperk\"ahler quotients. This criterion is applied to some…