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Anti-diagonal toric generalized K$\ddot{a}$hler structures of symplectic type on a compact toric symplectic manifold were investigated in \cite{Wang2} . In this article, we consider \emph{general} toric generalized K$\ddot{a}$hler…
In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost…
We extend our recent study of K3 metrics near the $T^4/Z_2$ orbifold locus to the other torus orbifold loci. In particular, we provide several new constructions of K3 surfaces as hyper-K\"ahler quotients, which yield new formulae for K3…
There is a well known one--parameter family of left invariant CR structures on $SU(2)\cong S^3$. We show how purely algebraic methods can be used to explicitly compute the canonical Cartan connections associated to these structures and…
We give a simple construction for hyperelliptic varieties defined as the quotient of a complex torus by the action of a dihedral group that contains no translations and fixes no points. This generalizes a construction given by Catanese and…
We study and classify almost complex totally geodesic submanifolds of the nearly Kaehler flag manifold $F_{1,2}(\mathbb C^3)$, and of its semi-Riemannian counterpart. We also develop a structural approach to the nearly Kaehler flag manifold…
We construct pseudotoric structures (\`{a} la Tyurin) on the two-step flag variety $F\ell_{1, n-1; n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important…
In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant…
This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let $X$ be an…
We provide the geometric construction of a series of generalized Schur algebras of any type via Borel-Moore homologies and equivariant K-groups of generalized Steinberg varieties. As applications, we obtain a Schur algebra analogue of the…
Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat K\"ahler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact…
The deformed double covering of E(2) group, denoted by $\tilde{E}_\kappa(2)$, is obtained by contraction from the $SU_\mu(2)$. The contraction procedure is then used for producing a new examples of differential calculi: 3D-left covariant…
The K-rings of non-singular complex pro jective varieties as well as quasi- toric manifolds were described in terms of generators and relations in an earlier work of the author with V. Uma. In this paper we obtain a similar description for…
In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds,…
We describe the geometry of the character variety of representations of the knot group $\Gamma_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally…
Given a symplectic manifold, we ask in how many different ways can a torus act on it. Classification theorems in equivariant symplectic geometry can sometimes tell that two Hamiltonian torus actions are inequivalent, but often they do not…
We provide a algebro-geometric combinatorial description of geometrically integral geometrically normal varieties endowed with an effective action of an algebraic torus over arbitrary fields. This description is achieved in terms of…
We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.
We construct new models of `curved' SU$(4|1)$ supersymmetric mechanics based on two versions of the off-shell multiplet ${\bf(8,8,0)}$ which are `mirror' to each other. The worldline realizations of the supergroup SU$(4|1)$ are treated as a…
Let G be a connected compact Lie group acting on a manifold M and let D be a transversally elliptic operator on M. The multiplicity of the index of D is a function on the set of irreducible representations of G. Let T be a maximal torus of…