arXiv subjects
Publications and source records attributed to Souradeep Majumder.
We study positivity properties of finite flat quotients of normal projective varieties. We show that the numerical groups and positive cones of these quotient varieties are related to those of the original varieties.
We compute the positive cones of the projectivization of a parabolic vector bundle and the fiber product of two parabolic projective bundles over a smooth complex projective curve. Specifically, we determine their Néron--Severi groups and compute their nef and pseudoeffective cones. Moreover, for the projectivization of a parabolic vector bundle, we explicitly describe the generators of the higher nef and pseudoeffective cones. As an application, we obtain a necessary and sufficient criterion for the semistability of a parabolic vector bundle.
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 $k$ be an algebraically closed field of any characteristic, and let $(X,P)$ be an orbifold curve over $k$. We construct the moduli space $\mathrm{M}_{(X,P)}^{\mathrm{ss}}(n, Δ)$ of $P$-semistable bundles on $(X,P)$ of rank $n$ and determinant $Δ$. In the characteristic zero case, this result is well known and follows from GIT techniques. Our construction follows a different approach inspired by a GIT-free construction of Faltings. We show that when the moduli space is non-empty, it is a finite disjoint union of irreducible projective varieties.
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 $C$ be a smooth, projective, geometrically irreducible curve defined over $\mathbb{R}$ such that $C(\mathbb{R}) = \emptyset$. Let $r>0$ and $d$ be integers which are coprime. Let $L$ be a line bundle on $C$ which corresponds to an $\mathbb{R}$ point of ${\rm Pic}^d_{C/\mathbb{R}}$. Let $\mathcal{M}_{r,L}$ be the moduli space of stable bundles on the complexification of $C$ of rank $r$ and determinant $L$. We classify birational types of $\mathcal{M}_{r,L}$ over $\mathbb{R}$.
Algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. It is shown that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined in \cite{KP}. Tensor, dual, pullback and pushforward operations are also defined for parabolic bundles.
We investigate principal bundles over a root stack. In case of dimension one, we generalize the criterion of Weil and Atiyah for a principal bundle to have an algebraic connection.
We investigate orthogonal and symplectic bundles with parabolic structure, over a curve.