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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 Kōta Yoshioka
Journal: J. Algebraic Geom. 20 (2011), 47-100
Published electronically: March 23, 2010
MathSciNet review: 2729275
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Abstract | Additional Information


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:

  1. [(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 $C$.

  2. [(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:

  1. [(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 $[C]$.

  2. [(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.

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
MR Author ID: 248505

Kōta Yoshioka
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Kobe 657-8501, Japan
MR Author ID: 352192

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