Kernel Sheaf on Integral Nodal Curves
In this article we study the stability of Kernel sheaf obtained from a generating subspace of rank one torsion-free sheaf on an integral nodal curve.
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arXiv subjects
Suratno Basu
Publications and source records attributed to Suratno Basu.
Existence of Ulrich Bundle on general Surfaces
Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle $K_X$. We further assume that $q -1 \le χ(\mathcal{O}_X$. In this article, we will study the existence of the Ulrich bundle and its stability property of it with respect to $K_X$.
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Generators of the cohomology ring, after Newstead
Newstead gave the generators of the cohomology ring of the moduli space of rank 2 semi-stable, torsion-free sheaves with fixed odd degree determinant over a smooth, projective curve. In this article, we generalize this result to the case when the underlying curve is irreducible, nodal. We show that these generators (of the cohomology ring in the nodal curve case) arise naturally as degeneration of Newstead's generators in the smooth curve case.
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A Torelli Type theorem for Nodal curves
The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.
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Stability of Syzygy bundles corresponding to stable vector bundles on algebraic surfaces
Let $(X, H)$ be a polarized smooth projective algebraic surface and $E$ is globally generated, stable vector bundle on $X$. Then the Syzygy bundle $M_E$ associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of $M_E$ with respect to $H$.
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Finite group actions on Higgs bundle moduli spaces
Let ${\cal M}(X,G)$ be the moduli space of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a semisimple complex Lie group with centre $Z$. We describe the fixed points of the action of a finite group $\Gamma$ on ${\cal M}(X,G)$, induced by holomorphic actions of $\Gamma$ on $X$ and $G$, a character of $\Gamma$ and a homomorphism from $\Gamma$ to the group of $Z$-bundles over $X$. Two important ingredients in this study are provided by the theory of twisted $\Gamma$-equivariant bundles developed by Barajas--Garc\'ia-Prada--Gothen--Mundet i Riera, and the Prym--Narasimhan--Ramanan construction given by Barajas--Garc\'ia-Prada. Via the non-abelian Hodge correspondence, our results provide a description of the fixed-point subvarieties of certain finite group actions on the $G$-character variety of the fundamental group of $X$.
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Degeneration of intermediate Jacobians and the Torelli theorem
Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed odd degree determinant. In this article we prove the analogous result in the case $X$ is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated Néron model of a family of such moduli spaces.
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System of Hodge Bundles and Generalized Opers on Smooth Projective Varieties
Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. We give criterion for semistability and stability of system of Hodge bundles on $X$. We define notion of generalized opers on $X$, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle $E$ together with a connection $\nabla$ over $X$ implies semistability of the pair $(E, \nabla)$.
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Stability of Secant Bundles on Second Symmetric Power of Curves
Given a rank $r$ stable bundle over a smooth irreducible projective curve $C,$ there is an associated rank $2r$ bundle over $S^2(C),$ the second symmetric power of $C.$ In this article we study the stability of this bundle. As a consequence we get an immersion from the moduli space of stable bundles over $C$ to the associated moduli space of stable bundles over $S^2(C).$
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Stability Of The Parabolic Poincaré Bundle
Given a compact Riemann surface $X$ and a moduli space $M_α(Λ)$ of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincaré parabolic bundle on $X\times M_α(Λ)$ is parabolic stable with respect to a natural polarization on $X\times M_α(Λ)$.
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On A Relative Mumford-Newstead Theorem
In this paper we prove a relative version of the classical Mumford-Newstead theorem for a family of smooth curves degenerating to a reducible curve with a simple node. We also prove a Torelli-type theorem by showing that certain moduli spaces of torsion-free sheaves on a reducible curve allows us to recover the curve from the moduli space.
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