In this paper, we introduce notions of $α$-planes in $5$D complex Heisenberg group and the twistor space as the moduli space of all $α$-planes. So we can define an anti-self-dual (ASD) connection as a connection flat over all $α$-planes. This geometric approach allows us to establish Penrose-Ward correspondence between ASD connections over $5$D complex Heisenberg group and a class of holomorphic vector bundles on the twistor space. By Atiyah-Ward ansätz, we also construct a family of ASD connections on $5$D complex Heisenberg group. When restricted to $5$D real Heisenberg group, the flat model of $5$D contact manifolds, an ASD connection satisfies the horizontal part of the contact instanton equation introduced by physicists.
We investigate the structure of a variety of new Moishezon twistor spaces, by utilizing the pluri-half-anti-canonical map from the twistor spaces. Each of these twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of ration…
The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel of the fibration. The heat kernel formula is explicit enough to derive small time asymptotics. Related twistor spaces and corresponding heat kernels are also discussed and the connection to the quaternionic magnetic Laplacian is done.
We describe the range of the Radon transform on the space $M$ of irreducible conics in $\CP^2$ in terms of natural differential operators associated to the $SO(3)$-structure on $M=SL(3, \R)/SO(3)$ and its complexification. Following \cite{moraru} we show that for any function $F$ in this range, the zero locus of $F$ is a four-manifold admitting an anti-self-dual conformal structure which contains three different scalar-flat Kähler metrics. The corresponding twistor space ${\mathcal Z}$ admits a holomorphic fibration over $\CP^2$. In the special case where ${\mathcal Z}=\CP^3\setminus\CP^1$ the…
We show that the algebraic dimension of a twistor space over n#CP^2 cannot be two if n>4 and the fundamental system (i.e. the linear system associated to the half-anti-canonical bundle, which is available on any twistor space) is a pencil. This means that if the algebraic dimension of a twistor space on n#CP^2, n>4, is two, then the fundamental system either is empty or consists of a single member. The existence problem for a twistor space on n#CP^2 with algebraic dimension two is open for n>4.
We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered.
Donaldson-Friedman constructed anti-self-dual classes on $K3\#3\overline{\mathbb{CP}_{2}}$ using twistor space. We show that some of these conformal classes have almost-Kähler representatives.
We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics: anti-self-duality (ASD) equations on Yang--Mills, or conformal curvature can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang--Mills, and …
A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $π$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $\mathbb{CP}^3$ of $S^4$. Deformations of twistor fibres project to real surfaces in $\mathbb{CP}^2$, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest t…
The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $ω$. This article studies limits of manifolds for which $ω$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorph…
