Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula


Authors: Hiraku Nakajima and Kota Yoshioka
Journal: J. Algebraic Geom. 20 (2011), 47-100
DOI: https://doi.org/10.1090/S1056-3911-10-00534-5
Published electronically: March 23, 2010
MathSciNet review: 2729275
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Abstract | References | Additional Information

Abstract: This is the second of series of papers studying moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up $ p\colon \widehat{X}\to X$ of a projective surface $ X$ at a point 0.

The main results of this paper are as follows:

(a)
We describe the wall-crossing between moduli spaces caused by the twisting of the line bundle $ \mathcal{O}(C)$ associated with the exceptional divisor $ C$.
(b)
We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves.

Moreover, we also give proofs which we observed in a special case in

[Perverse coherent sheaves on blow-up. I. A quiver description, preprint, arXiv:0802.3120] for the following:

(c)
The moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space of stable coherent sheaves on the original surface if the first Chern class is orthogonal to $ [C]$.
(d)
The moduli space becomes isomorphic to the usual moduli space of stable coherent sheaves on the blow-up after twisting by $ \mathcal{O}(-mC)$ for sufficiently large $ m$.
Therefore the usual moduli spaces of stable sheaves on the blow-up and the original surfaces are connected via wall-crossings.


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  • 1. E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, New York, 1985, xvi+386. MR 770932 (86h:14019)
  • 2. T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613-632. MR 1893007 (2003h:14027)
  • 3. T. Bridgeland and V. Toledano-Laredo, Stability conditions and Stokes factors, preprint, arXiv:0801.3974.
  • 4. J. Cheah, Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math. 183 (1998), no. 1, 39-90. MR 1616606 (99d:14002)
  • 5. L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193-207. MR 1032930 (91h:14007)
  • 6. D. Happel, I. Reiten and S.O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc., 120, (1996), no. 575, viii+88pp. MR 1327209 (97j:16009)
  • 7. M. He, Espaces de modules de systèmes cohérents, Internat. J. Math. 9 (1998), no. 5, 545-598. MR 1644040 (99i:14016)
  • 8. D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects of Math., E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870 (98g:14012)
  • 9. A. King, Instantons and holomorphic bundles on the blown up plane, Ph.D. thesis, Oxford, 1989.
  • 10. M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435.
  • 11. A. Langer, Moduli spaces of sheaves in mixed characteristic, Duke Math. J., 124 (2004), 571-586. MR 2085175 (2005g:14082)
  • 12. J. Li, Algebraic geometric interpretation of Donaldson's polynomial invariants, J. Differential Geom. 37 (1993), 417-466. MR 1205451 (93m:14007)
  • 13. I.G. Macdonald, Symmetric functions and Hall polynomials (2nd ed.), Oxford Math. Monographs, Oxford Univ. Press, 1995. MR 1354144 (96h:05207)
  • 14. E. Markman, Brill-Noether duality for moduli spaces of sheaves on K3 surfaces, J. Algebraic Geom. 10 (2001), 623-694. MR 1838974 (2002d:14065)
  • 15. M. Maruyama, Construction of moduli spaces of stable sheaves via Simpson's idea, in `Moduli of vector bundles' (Sanda, 1994; Kyoto, 1994), 147-187, Lecture Notes in Pure and Appl. Math., 179, Dekker, New York, 1996. MR 1397986 (97h:14020)
  • 16. K. Nagao, Derived categories of small toric Calabi-Yau 3-folds and counting invariants, preprint, arXiv:0809.2994.
  • 17. K. Nagao and H. Nakajima, Counting invariant of perverse coherent sheaves and its wall-crossing, preprint, arXiv:0809.2992.
  • 18. H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365-416. MR 1302318 (95i:53051)
  • 19. -, Quiver varieties and Kac-Moody algebras, Duke Math. 91 (1998), 515-560. MR 1604167 (99b:17033)
  • 20. -, Lectures on Hilbert schemes of points on surfaces, Univ. Lect. Ser. 18, AMS, 1999, xii+132 pp. MR 1711344 (2001b:14007)
  • 21. -, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), 671-721. MR 2023313 (2004k:16036)
  • 22. -, Convolution on homology groups of moduli spaces of sheaves on K3 surfaces, in ``Vector bundles and representation theory (Columbia, MO, 2002)'', 75-87, Contemp. Math. 322, 2003. MR 1987740 (2004e:14068)
  • 23. H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. $ 4$-dimensional pure gauge theory, Invent. Math 162 (2005), no. 2, 313-355. MR 2199008 (2007b:14027a)
  • 24. -, Lectures on instanton counting, Algebraic structures and moduli spaces, 31-101, CRM Proc. Lecture Notes, 38, Amer. Math. Soc., Providence, RI, 2004. MR 2095899 (2005m:14016)
  • 25. -, Perverse coherent sheaves on blow-up. I. A quiver description, preprint, arXiv:0802.3120.
  • 26. C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. I.H.E.S. 79 (1994), 47-129. MR 1307297 (96e:14012)
  • 27. M. Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691-723. MR 1333296 (96m:14017)
  • 28. M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423-455. MR 2057015 (2005e:14023)
  • 29. K. Yoshioka, Some examples of Mukai's reflections on K3 surfaces, J. Reine Angew. Math. 515 (1999), 97-123. MR 1717621 (2000h:14028)


Additional Information

Hiraku Nakajima
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Address at time of publication: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email: nakajima@math.kyoto-u.ac.jp

Kota Yoshioka
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Kobe 657-8501, Japan
Email: yoshioka@math.kobe-u.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-10-00534-5
Received by editor(s): June 27, 2008
Published electronically: March 23, 2010
Additional Notes: The first named author is supported by the Grant-in-aid for Scientific Research (No. 19340006), Japan Society for the Promotion of Science. A part of this work was done while the first named author was visiting the Institute for Advanced Study with supports by the Ministry of Education, Japan and the Friends of the Institute

American Mathematical Society