scholarly journals NC-smooth algebroid thickenings for families of vector bundles and quiver representations

2019 ◽  
Vol 155 (4) ◽  
pp. 681-710
Author(s):  
Ben Dyer ◽  
Alexander Polishchuk
Keyword(s):  
Deformation Quantization ◽  
Vector Bundles ◽  
Algebraic Varieties ◽  
Quiver Representations ◽  
Projective Curves

In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion ofalgebroidas a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations.

scholarly journals Spherical Hall Algebras of Weighted Projective Curves

2018 ◽  
Vol 2020 (15) ◽  
pp. 4721-4775
Author(s):  
Jyun-Ao Lin
Keyword(s):  
Vector Bundles ◽  
Characteristic Functions ◽  
Projective Curve ◽  
Hall Algebra ◽  
Smooth Projective Curve ◽  
Coherent Sheaves ◽  
Hall Algebras ◽  
Arbitrary Genus ◽  
Projective Curves ◽  
Fixed Rank

Abstract In this article, we deal with the structure of the spherical Hall algebra $\mathbf{U}$ of coherent sheaves with parabolic structures on a smooth projective curve $X$ of arbitrary genus $g$. We provide a shuffle-like presentation of the bundle part $\mathbf{U}^>$ and show the existence of generic spherical Hall algebra of genus $g$. We also prove that the algebra $\mathbf{U}$ contains the characteristic functions on all the Harder–Narasimhan strata. These results together imply Schiffmann’s theorem on the existence of Kac polynomials for parabolic vector bundles of fixed rank and multi-degree over $X$. On the other hand, the shuffle structure we obtain is new and we make links to the representations of quantum affine algebras of type $A$.

scholarly journals Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

2021 ◽  
Vol 20 (1) ◽  
pp. 95-119
Author(s):  
Nathan Grieve
Keyword(s):  
New York ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Index Theorem ◽  
Polynomial Functions ◽  
Algebraic Varieties ◽  
Wedderburn Decomposition ◽  
Generic Vanishing ◽  
Roch Theorem

Abstract We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves

2000 ◽  
Vol 43 (2) ◽  
pp. 129-137 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Stable Rank ◽  
Maximal Degree ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Projective Curves ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

AbstractLet E be a stable rank 2 vector bundle on a smooth projective curve X and V(E) be the set of all rank 1 subbundles of E with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, E, on X with fixed deg(E) and deg(L), L ∈ V(E) and such that .

scholarly journals Complex vector bundles on real algebraic varieties of small dimension

1988 ◽  
Vol 38 (3) ◽  
pp. 345-349
Author(s):  
Wojciech Kucharz
Keyword(s):  
Algebraic Variety ◽  
Vector Bundles ◽  
Small Dimension ◽  
Real Algebraic Variety ◽  
Algebraic Varieties ◽  
Complex Vector ◽  
Real Algebraic Varieties ◽  
Continuous Complex

LetXbe an affine real algebraic variety. In this paper, assuming that dimX≤ 7 and thatXsatisfies some other reasonable conditions, we give a characterisation of those continuous complex vector bundles onXwhich are topologically isomorphic to algebraic complex vector bundles onX.

scholarly journals ON THE DEFORMATION QUANTIZATION OF AFFINE ALGEBRAIC VARIETIES

2005 ◽  
Vol 16 (04) ◽  
pp. 419-435 ◽  
Author(s):  
R. FIORESI ◽  
M. A. LLEDÓ ◽  
V. S. VARADARAJAN
Keyword(s):  
Polynomial Ring ◽  
Poisson Structure ◽  
Deformation Quantization ◽  
Ambient Space ◽  
Algebraic Varieties ◽  
Algebraic Deformation ◽  
Poisson Variety

We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.

scholarly journals Moduli of vector bundles on higher-dimensional base manifolds — Construction and variation

2016 ◽  
Vol 27 (07) ◽  
pp. 1650054 ◽  
Author(s):  
Daniel Greb ◽  
Julius Ross ◽  
Matei Toma
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Recent Progress ◽  
Geometric Construction ◽  
Quiver Representations ◽  
Natural Morphism ◽  
Moduli Spaces Of Sheaves ◽  
Higher Dimensional ◽  
Moduli Of Vector Bundles

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson–Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker–Maruyama moduli spaces with respect to two different chosen polarizations are related via Thaddeus-flips through other “multi-Gieseker”-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson–Uhlenbeck moduli space.

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