Related papers: Higher order jet bundles of Lie group-valued funct…

The jet bundle $J^kG$ of $k$-jets of curves in a Lie group $G$ has a natural Lie group structure. We present an explicit formula for the group multiplication in the right trivialization and for the group 2-cocycle describing the abelian Lie…

We characterize the triples (X,L,H), consisting of holomorphic line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J_k(L), is isomorphic to a direct sum…

A Jet groupoid R_q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms from X to X. As a subbundle of the q-th order jet bundle of the trivial bundle X times X, a jet groupoid can be considered as a…

Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of…

Given a smooth plane quartic curve C over a field k of characteristic 0, with Jacobian variety J, and a marked rational point P of C(k), we construct a reductive group G and a G-variety X, together with an injection J(k)/2J(k) -> G(k)\X(k).…

We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms…

Let $X$ be the wonderful compactification of a complex adjoint symmetric space $G/K$ such that $rk(G/K)=rk(G)-rk(K)$. We show how to extend equivariant vector bundles on $G/K$ to equivariant vector bundles on $X$, generated by their global…

Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the…

Let $X$ be a differentiable manifold endowed with a transitive action $\alpha:A\times X\longrightarrow X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms…

A family of holomorphic vector bundles is constructed on a complex manifold $X$. The space of the holomorphic sections of these bundles are calculated in certain cases. As an application, if $X$ is an $N$-dimensional compact K\"ahler…

In this paper, we present the construction of a geometric object, called a generalized flag geometry, $(X^+;X^-)$, corresponding to a (2k +1)-graded Lie algebra $g=g_k\oplus\dots\oplus g_{-k}$. We prove that $(X^+;X^-) can be realized…

Given a Lie group G, one constructs a principal G-bundle on a manifold X by taking a cover U of X, specifying a transition cocycle on the cover, and descending the trivialized bundle along the cover. We demonstrate the existence of an…

Let G_2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a generic, smooth, connected, projective curve over $\mathbb{C}$ of genus at least 2. For a complex Lie group G, let H^0(M(G),L^k) be the space of…

The aim of this paper is to initiate a study of the jet bundles on the grassmannian $X$ over a field of characteristic zero using higher direct images of $G$-linearized sheaves, Lie theoretic methods, enveloping algebra theoretic methods…

Let $Y$ be a normal and projective variety over an algebraically closed field $k$ and $V$ a vector bundle over $Y$. We prove that if there exist a $k$-scheme $X$ and a finite surjective morphism $g:X\to Y$ that trivializes $V$ then $V$ is…

For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of $G$ (as defined by A. Connes) is shown to be a strict quantization (in the sense of M.…

Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $\mathsf{Bun}_{G,M}$ of principal…

Given a semisimple reductive group $G$ and a smooth projective curve $X$ over an algebraically closed field $k$ of arbitrary characteristic, let $\text{Bun}_G$ denote the moduli space of principal $G$-bundles over $X$. For a bundle…

Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of…

In this paper we generalize the theory of multiplicative $G$-Higgs bundles over a curve to pairs $(G,\theta)$, where $G$ is a reductive algebraic group and $\theta$ is an involution of $G$. This generalization involves the notion of a…