arXiv research

We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and the…

Real slices of parabolic opers on Riemann surfaces are studied.

New connections on symmetric spaces with invariant properties.

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is co…

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

We find all m-spin structures on Klein surfaces of genus larger than one. An m-spin structure on a Riemann surface P is a complex line bundle on P whose m-th tensor power is the cotangent bundle of P. A Klein surface can be described by a pair (P,tau), where P is a Riemann surface and tau is an anti-holomorphic involut…

Study real slices of SL(r,C)-opers via Riemann surface involution.

We introduce real structures on $L$-twisted Higgs pairs over a compact Riemann surface equipped with an anti-holomorphic involution, and prove a Hitchin--Kobayashi correspondence for them. Real $G$-Higgs bundles, where $G$ is a real form of a connected semisimple complex affine algebraic group $G^{\mathbb{C}}$, constit…

Let $Γ$ be a finitely generated group, $G_\mathbb{C}$ be a classical complex group and $G_\mathbb{R}$ a real form of $G_\mathbb{C}$. We propose a definition of the $G_\mathbb{R}$-character variety of $Γ$ as a subset $\mathcal{X}_{G_\mathbb{R}}(Γ)$ of the $G_\mathbb{C}$-character variety $\mathcal{X}_{G_\mathbb{C}}(Γ)$.…

A $k$-reflection of the $n$-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in ${\rm U}(n,1)$ with negative type eigenvalue $λ$, $|λ|=1$, of multiplicity $k+1$ and positive type eigenvalue $1$ of multiplicity $n-k$. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most fou…

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{Si…

Through the action of anti-holomorphic involutions on a compact Riemann surface, we construct families of (A,B,A)-branes in the moduli spaces of G_c-Higgs bundles on the Riemann surface. We study the geometry of these (A,B,A)-branes in terms of spectral data and show they have the structure of real integrable systems.

Study of algebraic curves and surfaces in flag manifold using twistor geometry.

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $σ$ such that $σ|_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric sp…

15 Einstein 4-manifolds with positive conformal curvature are classified.

Study gauge theory of real and quaternionic parabolic bundles over real curves.

We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(τ)\cap S$ of the fixed point set ${\rm fix}(τ)$ of an anti-holomorphic involution $τ$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ h…

New K3 surfaces with two involutions and low Picard number constructed.

Let $Γ$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $Γ$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $Γ$. We consider two anti-holomorphic involutions of the $\mathrm{SL}_n(\mathbb{C})$ character variet…

The paper studies spaces of non-compact real algebraic curves and their uniformisation.

Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution σ. Let G be a connected complex reductive affine algebraic group, and let σ_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to σand σ_G. We prove that the po…

We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP…

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, th…

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the o…

Study Higgs bundles on smooth projective varieties and their restrictions to curves.

The CR $δ$-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR $δ$-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic su…

It is well known that the twisters, section of twister space, classify the almost complex structure on even dimensional Riemannian manifold $X$. In this paper, it will be proved that a harmonic and anti-holomorphic twister is equivalent ti a symplectic structure on $X$.

This paper shows similarities in deformation spaces of Kleinian groups and anti-holomorphic maps.

Study of CR twistor model $Q^{2,2}$ and its sections.

Constructs stable bundles on K3 surfaces using monad construction.

The study of real Einstein submanifolds in Kähler geometry.

This paper solves the generalized Kähler problem by linking it to symplectic geometry.

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…

We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor $\mathcal{T}$ vanishes on the invariant vertical distribution. We give n…

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $Ω_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $φ: (Ω_1, d^K_{Ω_1}) \rightarrow (Ω_2, d^K_{Ω_2})$ is an isometry, i.e. $ d^K_…

It is shown that Einstein-Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev-Petviashvili (dKP) equation as a special case: If an EW structure admits a constant weighted vector then it is locally given by $h=d y^2-4d xd t-4ud t^2, ν=-4u_xd t$, where $u=u(x, y, t)$ satisfies the dKP equation $(u_…

Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and su…

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_G$. We define pseudo-real principal $G$--bundles on $X$; these are generalizations of re…

The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.

Classifies involutions on spherical 3-manifolds.

Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

New formula for dual knots using involutions.

The study classifies involutions on del Pezzo surfaces.

Characterizes the Legendre involution on generic frontals.

This paper introduces a complex representation for spacelike surfaces in the Lorentz-Minkowski space $L^4$, based in two complex valued functions which can be assumed to be holomorphic or anti-holomorphic. When the immersion is contained in quadrics of $L^4$, the representation then allows us to obtain interesting part…

The paper defines conditions for good involutions in generalized Alexander quandles.