1,158 research outputs found

    Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme

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    Let SS be an irreducible smooth projective surface defined over an algebraically closed field kk. For a positive integer dd, let Hilbd(S){\rm Hilb}^d(S) be the Hilbert scheme parametrizing the zero-dimensional subschemes of SS of length dd. For a vector bundle EE on SS, let H(E)β€‰βŸΆβ€‰Hilbd(S){\mathcal H}(E)\, \longrightarrow\, {\rm Hilb}^d(S) be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of SΓ—Hilbd(S)S\times {\rm Hilb}^d(S) as the kernel. We prove that two vector bundles EE and FF on SS are isomorphic if the vector bundles H(E){\mathcal H}(E) and H(F){\mathcal H}(F) are isomorphic.Comment: To appear in JRM

    Reconstructing vector bundles on curves from their direct image on symmetric powers

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    Let CC be an irreducible smooth complex projective curve, and let EE be an algebraic vector bundle of rank rr on CC. Associated to EE, there are vector bundles Fn(E){\mathcal F}_n(E) of rank nrnr on Sn(C)S^n(C), where Sn(C)S^n(C) is nβˆ’thsymmetricpowerof-th symmetric power of C.Weprovethefollowing:Let. We prove the following: Let E_1and and E_2betwosemistablevectorbundleson be two semistable vector bundles on C,with, with {\rm genus}(C)\, \geq\, 2.If. If {\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)forafixed for a fixed n,then, then E_1 \,=\, E_2$

    Hyperplane sections of projective bundle associated to the tangent bundle of P2.\mathbb{P}^2.

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    In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of P2\mathbb{P}^2 under its natural embedding in P7.\mathbb{P}^7.Comment: comments are welcome, revised version, some minor mistakes in the previous version is correcte

    Tangent bundle of \PP^2 and morphism from \PP^2 to \text{Gr}(2, \CC^{4})

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    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 H\mathcal{H} of the Hibert scheme of surfaces in \text{Gr}(2, \CC^{4}) with no point of it corresponds to a smooth surface.Comment: To appear in " proceedings of the march conference in Hyderabad.

    Equivariant vector bundles on complete symmetric varieties of minimal rank

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    Let XX be the wonderful compactification of a complex symmetric space G/HG/H of minimal rank. For a point xβ€‰βˆˆβ€‰Gx\,\in\, G, denote by ZZ be the closure of BxH/HBxH/H in XX, where BB is a Borel subgroup of GG. The universal cover of GG is denoted by G~\widetilde{G}. Given a G~\widetilde{G} equivariant vector bundle EE on X,X, we prove that EE is nef (respectively, ample) if and only if its restriction to ZZ is nef (respectively, ample). Similarly, EE is trivial if and only if its restriction to ZZ is so

    Automorphisms of Tβ€Ύ\overline{T}

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    Let Gβ€Ύ\overline G be the wonderful compactification of a simple affine algebraic group GG defined over C\mathbb C such that its center is trivial and G=ΜΈPSL(2,C)G\not= {\rm PSL}(2,\mathbb{C}). Take a maximal torus TβŠ‚GT \subset G, and denote by Tβ€Ύ\overline T its closure in Gβ€Ύ\overline G. We prove that TT coincides with the connected component, containing the identity element, of the group of automorphisms of the variety Tβ€Ύ\overline T.Comment: Final versio

    On equivariant principal bundles over wonderful compactifications

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    Let GG be a simple algebraic group of adjoint type over C\mathbb C, and let MM be the wonderful compactification of a symmetric space G/HG/H. Take a G~\widetilde G--equivariant principal RR--bundle EE on MM, where RR is a complex reductive algebraic group and G~\widetilde G is the universal cover of GG. If the action of the isotropy group H~\widetilde H on the fiber of EE at the identity coset is irreducible, then we prove that EE is polystable with respect to any polarization on MM. Further, for wonderful compactification of the quotient of PSL(n,C)\text{PSL}(n,{\mathbb C}), n ≠ 4n\,\neq\, 4 (respectively, PSL(2n,C)\text{PSL}(2n,{\mathbb C}), nβ‰₯2n \geq 2) by the normalizer of the projective orthogonal group (respectively, the projective symplectic group), we prove that the tangent bundle is stable with respect to any polarization on the wonderful compactification

    On a smooth compactification of PSL(n, C)/T

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    Let TT be a maximal torus of PSL(n,C){\rm PSL}(n, \mathbb C). For n β‰₯ 4n\,\geq\, 4, we construct a smooth compactification of PSL(n,C)/T{\rm PSL}(n, \mathbb C)/T as a geometric invariant theoretic quotient of the wonderful compactification PSL(n,C)β€Ύ\overline{{\rm PSL}(n, \mathbb C)} for a suitable choice of TT--linearized ample line bundle on PSL(n,C)β€Ύ\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 PSL(n,C)/T{\rm PSL}(n, \mathbb C)/T is PSL(n,C){\rm PSL}(n, \mathbb C) itself

    Positivity of vector bundles on homogeneous varieties

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    We study the following question: Given a vector bundle on a projective variety XX such that the restriction of EE to every closed curve Cβ€‰βŠ‚β€‰XC \,\subset\, X is ample, under what conditions EE is ample? We first consider the case of an abelian variety XX. If EE is a line bundle on XX, then we answer the question in the affirmative. When EE is of higher rank, we show that the answer is affirmative under some conditions on EE. We then study the case of X = G/PX \,=\, G/P, where GG is a reductive complex affine algebraic group, and PP is a parabolic subgroup of GG. In this case, we show that the answer to our question is affirmative if EE is TT--equivariant, where Tβ€‰βŠ‚β€‰PT\, \subset\, P is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on G/PG/P.Comment: Final version; 11 pages; to appear in International Journal of Mathematic

    A degeneration of moduli of Hitchin pairs

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    We construct a degeneration of the moduli space of Hitchin pairs on smooth projective curves when the curve degenerates to an irreducible curve with a single node. The degeneration constructed here is analogous to the models constructed by Gieseker and Nagaraj-Seshadri for the case of the usual moduli spaces (i.e when the Higgs structure is trivial). There is an canonical relative Hitchin map which is shown to be proper and the general fibre of the relative Hitchin map provides a new compactification of the Picard variety of smooth curves with normal crossing singularities
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