arXiv subjects
Publications and source records attributed to A. J. Parameswaran.
We explore the interconnections between the monodromy group of stratified bundles on a smooth projective variety $X$ and the monodromy of the strongly semistable vector bundles $V$ on $X$ such that $c_1(V)$ and $c_2(V)$ are numerically trivial.
We prove the existence of Ulrich bundles on cyclic coverings of $\mathbb{P}^n$ of arbitrary degree $d$. Given a relatively Ulrich bundle on a complete intersection subvariety, we construct a relatively Ulrich bundle on the ambient variety. As an application, we prove that there exists a rank $d$ Ulrich bundle on a generic cyclic covering of $\mathbb{P}^{2}$ of degree $d$, provided that the degree $d \cdot k$ of the branch divisor is even. When $d \cdot k$ is odd, we also provide an estimation of the rank of the Ulrich bundle on a generic cyclic covering of $\mathbb{P}^2$.
We compute the Harder-Narasimhan filtration of vector bundles $f_*\mathcal O_Y$ for certain finite morphisms $f\,:\,Y\,\longrightarrow\, X$ and in some other cases.
The notion of virtual global generation (VGG) for a vector bundle has multiple possible generalization from the case of curves to higher dimensional normal projective varieties. We study relationship between these notions. All these notions agree for curves but in higher dimension we show that this is not the case.
Let $H$ be a subgroup of ${\rm PGL}(2,\mathbb C)$ (respectively, ${\rm SL}(2,\mathbb C)$) such that the Zariski closure in ${\rm PGL}(2,\mathbb C)$ (respectively, ${\rm SL}(2,\mathbb C)$) of some compact subgroup of $H$ contains $H$. We classify the $H$--equivariant holomorphic vector bundles on $\mathbb{CP}^1$. This generalizes \cite{BM} where $H$ was assumed to be a finite abelian group.
We introduce three notion of tameness of the Nori fundamental group scheme for a normal quasiprojective variety $X$ over an algebraically closed field. It is proved that these three notions agree if $X$ admits a smooth completion with strict normal crossing divisor as the complement. We also prove a Lefschetz type restriction theorem for the tame Nori fundamental group scheme for such an $X$.
It is shown that if a finite generically smooth morphism $f\,:\,Y\,\longrightarrow\, X$ of smooth projective varieties induces an isomorphism of the étale fundamental groups, then the induced map of the stratified fundamental groups $π_1^{str}(f)\, :\, π_1^{str}(Y,\, y)\,\longrightarrow\, π_1^{str}(X,\, f(y))$ is also an isomorphism.
For any positive integer $k$, let $X_k$ be a projective irreducible nodal curve with $k$ nodes. We show that the Betti numbers and the mixed Hodge numbers of the compactified Jacobian $\overline{J_{k}}$ of an irreducible nodal curve $X_k$ with $k$ nodes are the same as the Betti numbers and the mixed Hodge numbers of $J_0\times R^k$, where $J_0$ is the Jacobian of the normalisation of the irreducible nodal curve and $R$ denotes the rational nodal curve with one node. We prove it by constructing a topologically locally trivial family of projective varieties containing $\overline{J_{k}}$ and $J_0\times R^k$ as fibres.
Let $f : X \rightarrow Y$ be a dominant generically smooth morphism between irreducible smooth projective curves over an algebraically closed field $k$ such that ${\rm Char}(k)> \text{degree}(f)$ if the characteristic of $k$ is nonzero. We prove that $(f_*{\mathcal O}_X)/{\mathcal O}_Y$ equipped with a natural parabolic structure is parabolic polystable. Several conditions are given that ensure that the parabolic vector bundle $(f_*{\mathcal O}_X)/{\mathcal O}_Y$ is actually parabolic stable.
Let $G$ be a connected semisimple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G.$ Let $F$ be an irreducible projective $B$-variety. Then consider the variety $E:=G\times^{B}F,$ which has a natural action of $G$; we call it $G$-induced variety or $(G,B)$-induced variety. In this article, we compute the connected component containing the identity automorphism of the group of all algebraic automorphisms of some particular $G$-induced varieties $E.$
Let $X \subset \mathbb P^3$ be a nonsingular cubic hypersurface. Faenzi (\cite{F}) and later Pons-Llopis and Tonini (\cite{PLT}) have completely characterized ACM line bundles over $X$. As a natural continuation of their study in the non-ACM direction, in this paper, we completely classify $\ell$-away ACM line bundles (introduced recently by Gawron and Genc (\cite{GG})) over $X$, when $\ell \leq 2$. For $\ell\geq 3$, we give examples of $\ell$-away ACM line bundles on $X$ and for each $\ell \geq 1$, we establish the existence of smooth hypersurfaces $X^{(d)}$ of degree $d >\ell$ in $\mathbb P^3$ admitting $\ell$-away ACM line bundles.
We prove Bertini type theorems and give some applications of them. The applications are in the context of Lefschetz theorem for Nori fundamental group for normal varieties as well as for geometric formal orbifolds. In another application, it is shown that certain class of a smooth quasi-projective variety contains a smooth curve such that irreducible lisse \ell-adic sheaves on the variety with "ramification bounded by a branch data" remains irreducible when restricted to the curve.
We give a characterization of genuinely ramified maps of formal orbifolds in the Tannakian framework. In particular we show that a morphism is genuinely ramified if and only if the pullback of every stable bundle remains stable in the orbifold category. We also give some other characterizations of genuine ramification. This generalizes the results of [BKP1] and [BP1]. In fact, it is a positive characteristic analogue of results in [BKP2].
Let $f:X\rightarrow Y$ be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle $((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is virtually globally generated. Moreover, $((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample if and only if $f$ is genuinely ramified.
The aim of this paper is to generalize the hyperplane section theorem of Gurjar to arbitrary (local) analytic varieties even if the intersection with of hyperplanes is not necessarily isolated. In case of formal varieties, we generalize the statement to work for different classes of functions than just hyperplanes. We call these classes (which are subsets of formal power series ring) to be algebraic $\mathbb{m}$-adicaly closed (A$\mathbb{m}$AC).
For any genuinely ramified morphism $f\, :\, Y\, \longrightarrow\, X$ between irreducible smooth projective curves we prove that $\overline{(Y\times_X Y) \setminus Δ}$ is connected, where $Δ\, \subset\, Y\times_X Y$ is the diagonal. Using this result the following are proved: If $f$ is further Morse then the Galois closure is the symmetric group $S_d$, where $d\,=\, \text{degree}(f)$. The Galois group of the general projection, to a line, of any smooth curve $X\,\subset\, \PP^n$ of degree $d$, which is not contained in a hyperplane and contains a non-flex point, is $S_d$.
Let $f : X \rightarrow Y$ be a generically smooth nonconstant morphism between irreducible projective curves, defined over an algebraically closed field, which is étale on an open subset of $Y$ that contains both the singular locus of $Y$ and the image, in $Y$, of the singular locus of $X$. We prove that the following statements are equivalent: \begin{enumerate} \item The homomorphism of étale fundamental groups $$f_* : π_1^{\rm et}(X) \rightarrowπ_1^{\rm et}(Y)$$ induced by $f$ is surjective. \item There is no nontrivial étale covering $ϕ: Y' \rightarrow Y$ admitting a morphism $q: X \rightarrow Y'$ such that $ϕ\circ q = f$. \item The fiber product $X\times_Y X$ is connected. \item $\dim H^0(X, f^*f_* {\mathcal O}_X)= 1$. \item ${\mathcal O}_Y \subset f_*{\mathcal O}_X$ is the maximal semistable subsheaf. \item The pullback $f^*E$ of every stable sheaf $E$ on $Y$ is also stable. \end{enumerate}
In this article, we prove that any smooth projective variety $X$ which is a double cover of the projective space $\mathbb{P}^n$ ($n\geq 2$) admits an Ulrich bundle. When $n=2$, we show that on any such $X$, there is an Ulrich bundle of rank two.