arXiv subjects
Publications and source records attributed to Sujoy Chakraborty.
Let $G$ be a connected affine algebraic group defined over a field of positive characteristic. We prove that the action of $G$ on a smooth projective variety can be lifted to its associated Artin-Schreier root stacks, whenever $G$ has no non-trivial characters. The existence of a $G$-linearization on a certain tautological invertible sheaf on such Artin-Schreier root stacks is also shown.
Let $C$ be an irreducible smooth projective curve of genus $g\geq 2$ over an algebraically closed field. We prove that the moduli stack of semi-stable vector bundles on $C$ of fixed rank and determinant is $\mathbb{A}^1$--connected. We also show that the moduli stack of quasi-parabolic vector bundles with a fixed determinant and a given quasi-parabolic data along a set of points in $C$ is $\mathbb{A}^1$-connected. Moreover, for small and generic weights $\boldsymbolα$ with $\gcd(n, °L) = 1$, the open substack of $\boldsymbolα$-semistable parabolic vector bundles is also $\mathbb{A}^1$-connected.
Let $X$ be a compact Riemann surface of genus at least $3$. We compute the Brauer groups of the moduli spaces of stable parabolic $\text{SL}(r,\mathbb{C})$-connections and stable strongly parabolic $\text{SL}(r,\mathbb{C})$-Higgs bundles over $X$. We also establish an equality of the Brauer group of the moduli stack of stable parabolic $\text{PGL}(r,\mathbb{C})$-connections and the smooth locus of its coarse moduli space.
We prove that the Brauer group of the moduli stack of parabolic stable principal $\text{PGL}(r,\mathbb{C})$-bundles on a curve $X$, for a generic system of weights along an arbitrary parabolic divisor, coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of parabolic stable principal $\text{PGL}(r,\mathbb{C})$-bundles. We also show that for any simple and simply connected complex linear algebraic group $G$, the analytic and algebraic Brauer groups of the moduli stack of quasi-parabolic principal $G$-bundles on $X$ vanish.
Let $X$ be a smooth connected complex projective curve of genus $g$, with $g\,\geq\, 3$. Fix an integer $r\geq 2$, a finite subset $D\, \subset\, X$, and a line bundle $L$ on $X$. We compute the Brauer group of the smooth locus of the moduli space of parabolic symplectic stable bundles of rank $r$ on $X$ equipped with a symplectic form taking values in $L(D)$, where $L(D)$ is given the trivial parabolic structure.
Take an irreducible smooth complex projective curve $X$ of genus $g$, with $g\,\geq\, 3$. Let $r$ be an even positive integer. We prove that the Brauer group of the moduli stack of stable parabolic $\textnormal{PSp}(r,\mathbb{C})$--bundles on $X$, of full-flag parabolic data along a set of marked points on $X$, coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of stable parabolic $\textnormal{PSp}(r,\mathbb{C})$--bundles. Under certain conditions on the parabolic types, we also compute the Brauer group of the smooth locus of this coarse moduli space. Similar computations are also done for the case of partial flags.
We propose a technique to protect and preserve a private key or a passcode in an encrypted two-dimensional graphical image. The plaintext private key or the passcode is converted into an encrypted QR code and embedded into a real-life color image with a steganographic scheme. The private key or the passcode is recovered from the stego color image by first extracting the encrypted QR code from the color image, followed by decryption of the QR code. The cryptographic key for encryption of the QR code is generated from the output of a Linear Feedback Shift Register (LFSR), initialized by a seed image chosen by the user. The user can store the seed image securely, without the knowledge of an attacker. Even if an active attacker modifies the seed image (without knowledge of the fact that it is the seed image), the user can easily restore it if he/she keeps multiple copies of it, so that the encryption key can be regenerated easily. Our experiments prove the feasibility of the technique using sample private key data and real-life color images.
Let $X$ be a smooth complex projective variety equipped with an action of a linear algebraic group $G$ over $\mathbb{C}$. Let $D$ be a reduced effective divisor on $X$ that is invariant under the $G$--action on $X$. Let $s_D$ be the canonical section of $\mathcal{O}_X(D)$ vanishing along $D$. Given a positive integer $r$, consider the stack $\mathfrak{X} := \mathfrak{X}_{(\mathcal{O}_X(D),\, s_D,\, r)}$ of $r$-th roots of $(\mathcal{O}_X, s_D)$ together with the natural morphism $π: \mathfrak{X} \to X$. Under the assumption that $G$ has no non-trivial characters, we show that the $G$--action on $X$ naturally lifts to a $G$--action on $\mathfrak{X}$ such that $π$ become $G$--equivariant, and the tautological invertible sheaf $\mathscr{M}$ on $\mathfrak{X}$ admits a linearization of this $G$--action. Finally, we define the notions of $G$--equivariant logarithmic connections on $\mathfrak{X}$ and $G$--equivariant parabolic connections on $X$ with rational parabolic weights along $D$, and establish an equivalence between the category of $G$--equivariant logarithmic connections on $\mathfrak{X}$ and the category of $G$--equivariant parabolic connections on $X$ with rational parabolic weights along $D$.
We consider the moduli space of stable parabolic Higgs bundles of rank $r$ and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth complex projective curve $X$ of genus $g$, with $g\,\geq\, 2$. The group $Γ$ of $r$-torsion points of the Jacobian of $X$ acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from $Γ$. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen--Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen--Ruan product structure in special cases.
Let $D$ be a reduced effective strict normal crossing divisor on a smooth complex variety $X$, and let $\mathfrak{X}_D$ be an associated root stack over $\mathbb C$. Suppose that $X$ admits an anti-holomorphic involution (real structure) that keeps $D$ invariant. We show that the root stack $\mathfrak{X}_D$ naturally admits a real structure compatible with $X$. We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on $X$.
Let $D$ be an effective divisor on a smooth projective variety $X$ over an algebraically closed field $k$ of characteristic $0$. We show that there is a one-to-one correspondence between the class of orthogonal (respectively, symplectic) parabolic vector bundles on $X$ with parabolic structure along $D$ and having rational weights and the class of orthogonal (respectively, symplectic) vector bundles on certain root stacks associated to this data. Using this, we describe the orthogonal (respectively, symplectic) vector bundles on the root stack as reductions of the structure group to orthogonal (respectively, symplectic) groups. When $D$ is a divisor with strict normal crossings, we prove a one-to-one correspondence between the class of orthogonal (respectively, symplectic) parabolic connections on $X$ with rational weights, and the class of orthogonal (respectively, symplectic) logarithmic connections on certain fiber product of root stacks with poles along a divisor with strict normal crossings.
Let $k$ be an algebraically closed field of characteristic zero. We prove that the Brauer group of moduli stack of stable parabolic $\textnormal{PGL}(r,k)$-bundles with full flag quasi-parabolic structures at an arbitrary parabolic divisor on a curve $X$ coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of parabolic $\textnormal{PGL}(r,k)$-bundles. We also compute the Brauer group of the smooth locus of this coarse moduli for more general quasi-parabolic types and weights satisfying certain conditions.
We study the Chow group of 1-cycles of the moduli space of semistable parabolic vector bundles of fixed rank, determinant and a generic weight over a nonsingular projective curve over $\mathbb{C}$ of genus at least 3. We show that, the Chow group of 1-cycles remains isomorphic as we vary the generic weight. As a consequence, we can give an explicit description of the Chow group in the case of rank 2 and determinant $\mathcal{O}(x)$, where $x\in X$ is a fixed point.
Let $C$ be a nonsingular complex projective curve, and $\mathcal{L}$ e a line bundle of degree 1 on $C$. Let $\mathcal{M}_α := \mathcal{M}(r,\mathcal{L},α)$ denote the moduli space of $S$-equivalence classes of Parabolic stable bundles of fixed rank $r$, determinant $\mathcal{L}$, full flags and generic weight $α$. Let $n=$ dim$\mathcal{M}_α$. We aim to study the Abel-Jacobi maps for $\mathcal{M}_α$ in the cases $k=2,n-1$. When $k=n-1$, we prove that the Abel-Jacobi map is a split surjection. When $k=2$ and $r=2$, we show that the Abel-Jacobi map is an isomorphism.
Let $X$ be an irreducible smooth projective curve of genus $g \geq 2$ over $\mathbb{C}$. Let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$. Let $\mathrm{M}_{G, {\rm Higgs}}^δ$ be the moduli space of semistable principal $G$--Higgs bundles on $X$ of topological type $δ\in π_1(G)$. In this article, we compute the fundamental group and Picard group of $\mathrm{M}_{G, {\rm Higgs}}^δ$.