arXiv subjects
Publications and source records attributed to D. S. Nagaraj.
On the flag variety $ \mathcal{F}l_s(E)$ associated to a vector bundle $E,$ , a sequence $s$ and a partition $a,$ there is a line bundle $\it Q^a_s$ on $ \mathcal{F}l_s(E).$ The aim of this paper is to prove the following conjecture: $Q^a_s $ on $ \mathcal{F}l_s(E)$ is big if only if $π_*(Q^a_s)=S_a(E)$ on X is big. The "if" part is proven here, the "only if" part is proven under the V-bigness hypothesis.
In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary conditions for existence of determinantal representation. As an application, we show that for any integer $d \geq 1,$ there is an indecomposable vector bundle $E_d$ of rank $2$ on $\mathbb{P}^2$ such that almost all curves of degree $d$ of $\mathbb{P}^2$ arise as the degeneracy loci of a pair of holomorphic sections of $E_d$, upto an automorphism of $\mathbb{P}^2$. We use this result to obtain a linear algebraic application.
We prove that a vector bundle $E$ over a smooth complex projective variety $M$ is étale trivial if and only if $E$ is semiample and $c_1(E) \in H^2(M, {\mathbb Q})$ vanishes. Also, a vector bundle $E$ over a smooth complex projective curve is semiample if and only if it is virtually globally generated.
We prove that the direct image of an anti-ample vector bundle is anti-ample under any finite flat morphism of non-singular projective varieties. In the second part we prove some properties of big and nef vector bundles. In particular it is shown that the tensor product of a nef vector bundle with a nef and big vector bundle is again nef and big. This generalizes a result of Schneider.
We give an algebraic-geometric proof of the fact that for a smooth fibration $π: X \longrightarrow Y$ of projective varieties, the direct image $π_*(L\otimes K_{X/Y})$ of the adjoint line bundle of an ample (respectively, nef and $π$-strongly big) line bundle $L$ is ample (respectively, nef and big).
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 $\mathbb{P}^7.$ As an application one obtains a description of all possible deformations, in $\text{SL_3(\mathbb{C}}/B,$ of the co-dimension one sub-scheme which is the union of two fundamental Schubert divisors.
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 $G/P$.
The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.
Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$, let $\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of Seshadri constant. Our results generalize various known results for ${\rm rank}(E)=2$.
This is a survey article on recent results on vector bundles on symmetric product of non-singular projective curves.
This is a survey article about parabolic bundles and parabolic Higgs bundles.
In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2 vector bundles $E$ on some specific type of $n$ dimensional projective manifold $X.$ The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.
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.
In this note we describe the image of $\PP^2$ in $ Gr(2, \CC^{4})$ under a morphism given by a rank two vector bundle on $\PP^2$ with Chern classes $(2,2).$
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.
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 $\overline T$.
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.
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.