Related papers: Toric anti-self-dual 4-manifolds via complex geome…

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set…

This is a continuation of arXiv: 2408.03012. We answer affirmatively Question 5.10 posed in the previous article. More precisely, let $(X, \omega)$ be a conical symplectic variety of dimension $2n$ with $wt(\omega) = 2$, which has a…

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties,…

We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.

We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup W, of the isometry group of the weight 2 Hodge structure,…

Let a torus act on a compact oriented manifold $M$ with isolated fixed points, with an additional mild assumption that its isotropy submanifolds are orientable. We associate a signed labeled multigraph encoding the fixed point data (weights…

Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus…

We perform a topological-holomorphic twist of $\mathcal{N}=4$ supersymmetric gauge theory on a four-manifold of the form $M_4=\Sigma_1 \times \Sigma_2$ with Riemann surfaces $\Sigma_{1,2}$, and unravel the mathematical implications of its…

An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a…

As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex…

We study general linear perturbations of a class of 4d real-dimensional hyperkahler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic…

We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by…

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary…

Ingoing and outgoing principal null geodesics in Kerr spacetimes are characterized as part of parametrized families of strings in complex Kerr geometry and are associated with holomorphic curves in twistor space with help of the Kerr…

We consider the classification of supersymmetric black hole solutions to five-dimensional STU gauged supergravity that admit torus symmetry. This reduces to a problem in toric K\"ahler geometry on the base space. We introduce the class of…

We show the existence of strictly almost-Kahler anti-self-dual metrics on certain 4-manifolds by deforming scalar-flat Kahler metrics. On the other hand, we prove the non-existence of such metrics on certain other 4-manifolds by means of…

In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold $(X,L)$, we choose a maximal algebraic torus $T$ in the group of holomorphic…

We briefly review the hierarchy for the hyper-K\"ahler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy with a…

A twist is a datum playing a role of a local system for topological $K$-theory. In equivariant setting, twists are classified into four types according to how they are realized geometrically. This paper lists the possible types of twists…

Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR 5-manifolds defined as $\mathbb{P}^1$-bundles over 3-dimensional conformal manifolds. In this paper, we embed a real analytic twistor CR manifold into the twistor space…