1,158 research outputs found

### Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme

Let $S$ be an irreducible smooth projective surface defined over an
algebraically closed field $k$. For a positive integer $d$, let ${\rm
Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes
of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\mathcal H}(E)\,
\longrightarrow\, {\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed
using the structure sheaf of the universal subscheme of $S\times {\rm
Hilb}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$
are isomorphic if the vector bundles ${\mathcal H}(E)$ and ${\mathcal H}(F)$
are isomorphic.Comment: To appear in JRM

### Reconstructing vector bundles on curves from their direct image on symmetric powers

Let $C$ be an irreducible smooth complex projective curve, and let $E$ be an
algebraic vector bundle of rank $r$ on $C$. Associated to $E$, there are vector
bundles ${\mathcal F}_n(E)$ of rank $nr$ on $S^n(C)$, where $S^n(C)$ is n$-th symmetric power of$C$. We prove the following: Let$E_1$and$E_2$be
two semistable vector bundles on$C$, with${\rm genus}(C)\, \geq\, 2$. If${\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)$for a fixed$n$, then$E_1
\,=\, E_2$

### Hyperplane sections of projective bundle associated to the tangent bundle of $\mathbb{P}^2.$

In this note we give a complete description of all the hyperplane section of
the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under
its natural embedding in $\mathbb{P}^7.$Comment: comments are welcome, revised version, some minor mistakes in the
previous version is correcte

### Tangent bundle of \PP^2 and morphism from \PP^2 to \text{Gr}(2, \CC^{4})

In this note we study the image of \PP^2 in \text{Gr}(2, \CC^{4}) given
by tangent bundle of \PP^2. We show that there is component $\mathcal{H}$ of
the Hibert scheme of surfaces in \text{Gr}(2, \CC^{4}) with no point of it
corresponds to a smooth surface.Comment: To appear in " proceedings of the march conference in Hyderabad.

### Equivariant vector bundles on complete symmetric varieties of minimal rank

Let $X$ be the wonderful compactification of a complex symmetric space $G/H$
of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of
$BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of
$G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector
bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only
if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is
trivial if and only if its restriction to $Z$ is so

### Automorphisms of $\overline{T}$

Let $\overline G$ be the wonderful compactification of a simple affine
algebraic group $G$ defined over $\mathbb C$ such that its center is trivial
and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and
denote by $\overline T$ its closure in $\overline G$. We prove that $T$
coincides with the connected component, containing the identity element, of the
group of automorphisms of the variety $\overline T$.Comment: Final versio

### On equivariant principal bundles over wonderful compactifications

Let $G$ be a simple algebraic group of adjoint type over $\mathbb C$, and let
$M$ be the wonderful compactification of a symmetric space $G/H$. Take a
$\widetilde G$--equivariant principal $R$--bundle $E$ on $M$, where $R$ is a
complex reductive algebraic group and $\widetilde G$ is the universal cover of
$G$. If the action of the isotropy group $\widetilde H$ on the fiber of $E$ at
the identity coset is irreducible, then we prove that $E$ is polystable with
respect to any polarization on $M$. Further, for wonderful compactification of
the quotient of $\text{PSL}(n,{\mathbb C})$, $n\,\neq\, 4$ (respectively,
$\text{PSL}(2n,{\mathbb C})$, $n \geq 2$) by the normalizer of the projective
orthogonal group (respectively, the projective symplectic group), we prove that
the tangent bundle is stable with respect to any polarization on the wonderful
compactification

### On a smooth compactification of PSL(n, C)/T

Let $T$ be a maximal torus of ${\rm PSL}(n, \mathbb C)$. For $n\,\geq\, 4$,
we construct a smooth compactification of ${\rm PSL}(n, \mathbb C)/T$ as a
geometric invariant theoretic quotient of the wonderful compactification
$\overline{{\rm PSL}(n, \mathbb C)}$ for a suitable choice of $T$--linearized
ample line bundle on $\overline{{\rm PSL}(n, \mathbb C)}$. We also prove that
the connected component, containing the identity element, of the automorphism
group of this compactification of ${\rm PSL}(n, \mathbb C)/T$ is ${\rm PSL}(n,
\mathbb C)$ itself

### Positivity of vector bundles on homogeneous varieties

We study the following question: Given a vector bundle on a projective
variety $X$ such that the restriction of $E$ to every closed curve $C
\,\subset\, X$ is ample, under what conditions $E$ is ample? We first consider
the case of an abelian variety $X$. If $E$ is a line bundle on $X$, then we
answer the question in the affirmative. When $E$ is of higher rank, we show
that the answer is affirmative under some conditions on $E$. We then study the
case of $X \,=\, G/P$, where $G$ is a reductive complex affine algebraic group,
and $P$ is a parabolic subgroup of $G$. In this case, we show that the answer
to our question is affirmative if $E$ is $T$--equivariant, where $T\, \subset\,
P$ is a fixed maximal torus. Finally, we compute the Seshadri constant for such
vector bundles defined on $G/P$.Comment: Final version; 11 pages; to appear in International Journal of
Mathematic

### A degeneration of moduli of Hitchin pairs

We construct a degeneration of the moduli space of Hitchin pairs on smooth
projective curves when the curve degenerates to an irreducible curve with a
single node. The degeneration constructed here is analogous to the models
constructed by Gieseker and Nagaraj-Seshadri for the case of the usual moduli
spaces (i.e when the Higgs structure is trivial). There is an canonical
relative Hitchin map which is shown to be proper and the general fibre of the
relative Hitchin map provides a new compactification of the Picard variety of
smooth curves with normal crossing singularities

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