Related papers: Secondary staircase complexes on isotropic Grassma…

For any (n-1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the…

We present a construction (and classification) of certain invariant 2-forms on the real symplectic group. They are used to define a symplectic form on the quotient by a maximal torus and to "lift" a symplectic structure from a symplectic…

We present an explicit construction of tilting bundles on cotangent bundles of Grassmannians of 2-planes. This construction is based on Kapranov's exceptional collection for the underlying Grassmannians, and utilizes specific iterative…

Let X be a complex projective curve which is smooth and irreducible of genus 2. The moduli space M_2 of semistable symplectic vector bundles of rank 4 over X is a variety of dimension 10. After assembling some results on vector bundles of…

We present a class of symplectic matrices which transform by similarity given $2n\times 2n$ -dimensional matrix into Bunse-Gerstner form. If the given matrix is skew-Hamiltonian, the transformation gives a solution of an antisymmetric…

In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal…

In 2017, Catanese--Perroni gave a natural correspondence between the Picard group of a double cover and a set of pairs of a vector bundle of rank two and a certain morphism of vector bundles on the base space. In this paper, we describe the…

We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $\Phi$ from the fan of…

In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal…

Let E be the total space of a locally trivial torus bundle over the surface \Sigma_g of genus g>1. Using the Seiberg--Witten theory and spectral sequences we prove that E carries a symplectic structure if and only if the homology class of…

Tractors and Twistors bundles both provide natural conformally covariant calculi on $4D$-Riemannian manifolds. They have different origins but are closely related, and usually constructed bottom-up through prolongation of defining…

The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq…

We present a survey on the moduli spaces of rank 2 quadric bundles over a compact Riemann surface X. These are objects which generalise orthogonal bundles and which naturally occur through the study of the connected components of the moduli…

In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. We give self-contained proofs here. One often encounters these…

Symplectic torus bundles $\xi:T^{2}\to E\to B$ are classified by the second cohomology group of $B$ with local coefficients $H_{1}(T^{2})$. For $B$ a compact, orientable surface, the main theorem of this paper gives a necessary and…

The aim of this paper is two-fold. First, we define symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. We call these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs…

Let $X$ be a submanifold of dimension $n$ of the complex projective space $\mathbb P^N$ ($n<N$), and let $E$ be a vector bundle of rank two on $X$ . If $n\geq\frac{N+3}{2}\geq 4$ we prove a geometric criterion for the existence of an…

Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of…

In this paper, we describe the category of bi-equivariant vector bundles on a bi-equivariant smooth (partial) compactification of a reductive algebraic group with normal crossing boundary divisors. Our result is a generalization of the…

We give another proof of the generic semisimplicity of the big quantum cohomology of the symplectic isotropic Grassmannians IG(2,2n).