Related papers: Some results on equivariant contact geometry for p…

We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type $A_{n-1}$. This is obtained inside the limit of a projective system of Borel-Moore homologies of…

Atiyah's formulation of what is nowadays called the convexity theorem of Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map arising from a Hamiltonian action of a torus on a symplectic manifold is a convex polytope,…

We prove sign-alternation of the structure constants in the basis of structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the flag varieties $G/P$ associated to an arbitrary…

It was recently shown (by the second author and D\'{i}az Garc\'{i}a, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant…

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…

We investigate the local geometry of a pair of independent contact structures on 3-manifolds under maps that independently preserve each contact structure. We discover that such maps are homotheties on the contact 1-forms and we discover…

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it…

We apply a theorem of Gel'fand, Goresky, MacPherson, and Serganova about matroid polytopes to study semistability of partial flags relative to a T-linearized ample line bundle of a flag space F = SL(n)/P where T is a maximal torus in SL(n)…

Let $ G $ be a connected reductive algebraic group over $ \mathbb{R} $, and $ H $ its symmetric subgroup. For parabolic subgroups $ P_{G} \subset G $ and $ P_{H} \subset H $, the product of flag varieties $ \mathfrak{X} = H/P_H \times G/P_G…

We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic…

In this paper we give a geometric construction of the Borel equivariant (co)homology for spaces with a $G$-action, where $G$ is a compact Lie group with the property that the adjoint representation is orientable. A nice feature of these…

We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong…

Let X be a smooth complete complex toric variety such that the boundary is a simple normal crossing divisor, and let E be a holomorphic vector bundle on X. We prove that E admits an equivariant structure if and only if E admits a…

Let $\XR$ be a (generalized) flag manifold of a non-compact real semisimple Lie group $\GR$, where $\XR$ and $\GR$ have complexifications X and G. We investigate the problem of constructing a graded star product on $Pol(T^*\XR)$ which…

We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on…

Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic…

We show that local complete intersection subvarieties of smooth projective varieties, which have partially ample normal bundle, possess the G2-property. This generalizes results of Hartshorne and B\u{a}descu-Schneider.

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over…

This short note considers varieties of the form $G\times S_{\text{reg}}$, where $G$ is a complex semisimple group and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. Such varieties arise naturally in hyperk\"ahler…

For unramified reductive groups, we determine the connected components of affine Deligne-Lusztig varieties in the partial affine flag varieties. Based on the work of Hamacher-Kim and Zhou, this result allows us to verify, in the unramified…