SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
We show that the characteristic function 1<수식> of any Harder-Narasimhan strata <수식> belongs to the spherical Hall algebra <수식> of a smooth projective curve X (defined over a finite field Fq). We prove a similar result in the geometric setting: the intersection coho-mology complex IC(<수식>) of any Harder-Narasimhan strata <수식>belongs to the category QX of spherical Eisenstein sheaves of X. We show by a simple example how a complete description of all spherical Eisenstein sheaves would necessarily involve the Brill-Noether stratas of <수식>.
Abstract
We show that the characteristic function 1<수식> of any Harder-Narasimhan strata <수식> belongs to the spherical Hall algebra <수식> of a smooth projective curve X (defined over a finite field Fq). We prove a similar result in the geometric setting: the intersection coho-mology complex IC(<수식>) of any Harder-Narasimhan strata <수식>belongs to the category QX of spherical Eisenstein sheaves of X. We show by a simple example how a complete description of all spherical Eisenstein sheaves would necessarily involve the Brill-Noether stratas of <수식>.
참고문헌 (21)
- E. Arbarello, "Geometry of algebraic curves. Volume I, Grundlehren Math. Wiss. 267" (1985), Springer-Verlag
- P. Baumann, "The Hall algebra of the category of coherent sheaves on the projective line", J. Reine Angew. Math, 533, pp.207~233 (2001)
- A. Beilinson, "Faisceaux pervers, Analysis and topology on singular spaces", Asterisque, 100, pp.5~171 (1982), Soc. Math
- 0505148, to appear in Duke Math. Jour, "On the Hall algebra of an elliptic curve, I., preprint math.AG", I. Burban
- G. Harder, "On the cohomology groups of moduli spaces of vector bundles on curves", Math. Ann, 212, pp.215~248 (1975)
- M. Kapranov, "Eisenstein series and quantum affine algebras, Algebraic geometry 7", J. Math. Sci, 84(5), pp.1311~1360 (1997)
- M. Kashiwara, "Geometric construction of crystal bases", Duke Math. J, 89(1), pp.9~36 (1997)
- "A survey of Brill-Noether theory on algebraic curves" (1993)[URL]
- G. Laumon, "Faisceaux automorphes lies aux series d'Eisenstein, Automorphic forms, Shimura varieties, and L-functions, Vol. I", Perspect. Math, 10, pp.227~281 (1990), Academic Press
- G. Lusztig, "Introduction to Quantum Groups" (1994), Birkhauser
- H. Nakajima, "Quiver varieties and Kac-Moody algebras", Duke Math. J, 91(3), pp.515~560 (1998)
- M. Reineke, "The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli", Invent. Math, 152(2), pp.349~368 (2003)
- C. Ringel, "Hall algebras and quantum groups", Invent. Math, 101(3), pp.583~591 (1990)
- O. Schiffmann, "Noncommutative projective curves and quantum loop algebras", Duke Math. J, 121(1), pp.113~168 (2004)
- O. Schiffmann, "Lectures on Hall algebras", arXiv:math, 2006
- O. Schiffmann, "Canonical bases and moduli spaces of sheaves on curves", Invent. Math, 165(3), pp.453~524 (2006)
- O. Schiffmann, "On the Hall algebra of an elliptic curve, II", preprint arXiv:math, 2005
- O. Schiffmann, "The elliptic Hall algebra, Cherednick Hecke algebras and Macdonald polynomials", Compos. Math, 147(1), pp.188~234 (2011)
- O. Schiffmann, "Hall algebras of curves, quiver varieties and Langlands duality", arXiv:1009.0678, ~ (2010)
- S. Shatz, "The decomposition and specialization of algebraic families of vector bundles", Compositio Math, 35(2), pp.163~187 (1977)
- M. Varagnolo, "On the decomposition matrices of the quantized Schur algebra", Duke Math. J, 100(2), pp.267~297 (1999)