Kyoto Journal of Mathematics

Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing

Hiraku Nakajima and Kōta Yoshioka

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Abstract

In earlier papers of this series we constructed a sequence of intermediate moduli spaces $\{{\widehat{M}}^{m}(c)\}_{m=0,1,2,\ldots}$ connecting a moduli space $M(c)$ of stable torsion-free sheaves on a nonsingular complex projective surface $X$ and ${\widehat{M}}(c)$ on its one-point blow-up $\widehat {X}$. They are moduli spaces of perverse coherent sheaves on $\widehat{X}$. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from ${\widehat{M}}^{m}(c)$ to ${\widehat{M}}^{m+1}(c)$ and then from $M(c)$ to ${\widehat{M}}(c)$. As an application we prove that Nekrasov-type partition functions satisfy certain equations that determine invariants recursively in second Chern classes. They are generalizations of the blow-up equation for the original Nekrasov deformed partition function for the pure $N=2$ supersymmetric gauge theory, found and used to derive the Seiberg-Witten curves.

Article information

Source
Kyoto J. Math. Volume 51, Number 2 (2011), 263-335.

Dates
First available: 22 April 2011

Permanent link to this document
http://projecteuclid.org/euclid.kjm/1303494505

Digital Object Identifier
doi:10.1215/21562261-1214366

Zentralblatt MATH identifier
05907268

Mathematical Reviews number (MathSciNet)
MR2793270

Subjects
Primary: 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) [See also 32L25, 81Txx]
Secondary: 16G20: Representations of quivers and partially ordered sets

Citation

Nakajima, Hiraku; Yoshioka, Kōta. Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing. Kyoto Journal of Mathematics 51 (2011), no. 2, 263--335. doi:10.1215/21562261-1214366. http://projecteuclid.org/euclid.kjm/1303494505.


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References

  • [1] R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248.
  • [2] A. Braverman and P. Etingof, “Instanton counting via affine Lie algebras, II: From Whittaker vectors to the Seiberg-Witten prepotential” in Studies in Lie Theory, Progr. Math. 243, Birkhäuser, Boston, 2006, 61–78.
  • [3] G. Ellingsrud and L. Göttsche, Variation of moduli spaces and Donaldson invariants under change of polarization, J. Reine Angew. Math. 467 (1995), 1–49.
  • [4] J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Math. 352, Springer, Berlin, 1973.
  • [5] W. Fulton and S. Lang, Riemann-Roch Algebra, Grundlehren Math. Wiss. 277, Springer, New York, 1985.
  • [6] A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov, RG equations from Whitham hierarchy, Nuclear Phys. B 527 (1998), 690–716.
  • [7] L. Göttsche, H. Nakajima, and K. Yoshioka, Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008), 343–390.
  • [8] L. Göttsche, H. Nakajima, and K. Yoshioka, K-theoretic Donaldson invariants via instanton counting, Pure Appl. Math. 5 (2009), 1029–1111.
  • [9] R. Joshua, Equivariant Riemann-Roch for G-quasi-projective varieties, I, K-Theory 17 (1999), 1–35.
  • [10] A. King, Moduli of representations of finite dimensional algebras, Quart. J. Oxford Ser. Math. (2) 45 (1994), 515–530.
  • [11] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
  • [12] A. Losev, N. Nekrasov, and S. Shatashvili, Issues in topological gauge theory, Nuclear Phys. B 534 (1998), 549–611.
  • [13] A. Marshakov and N. Nekrasov, Extended Seiberg-Witten theory and integrable hierarchy, J. High Energy Phys. 2007, no. 1, art. id. 104.
  • [14] T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition of Moduli Stacks, Lecture Notes in Math. 1972, Springer, Berlin, 2009.
  • [15] D. Mumford, Tata lectures on theta. II: Jacobian Theta Functions and Differential Equations, reprint of the 1984 original, Mod. Birkhäuser Class., Birkhäuser, Boston, 2007.
  • [16] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
  • [17] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, Univ. Lect. Ser. 18, Amer. Math. Soc., Providence, 1999.
  • [18] H. Nakajima and K. Yoshioka, “Lectures on instanton counting” in Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes 38, Amer. Math. Soc., Providence, 2004, 31–101.
  • [19] H. Nakajima and K. Yoshioka, Instanton counting on blowup, I: 4-dimensional pure gauge theory, Invent. Math. 162 (2005), 313–355.
  • [20] H. Nakajima and K. Yoshioka, Instanton counting on blowup, II: K-theoretic partition function, Transform. Groups 10 (2005), 489–519.
  • [21] H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up, II: Wall-crossing and Betti numbers formula, J. Algebraic Geom. 29 (2011), 47–100.
  • [22] H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up, I: A quiver description, preprint, arXiv:0802.3120v2 [math.AG]
  • [23] H. Nakajima and K. Yoshioka, Instanton counting on blowup, III: Theories with matters, in preparation.
  • [24] N. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), 831–864.
  • [25] N. Nekrasov and A. Okounkov, “Seiberg-Witten prepotential and random partitions” in The Unity of Mathematics, Progr. Math. 244, Birkhäuser, Boston, 2006, 525–596.
  • [26] Y. Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, J. High Energy Phys. 2004, no. 2, art. id. 050.
  • [27] M. Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), 691–723.
  • [28] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.
  • [29] K. Yamada, Blowing-ups describing the polarization change of moduli schemes of semistable sheaves of general rank, Comm. Algebra 38 (2010), 3094–3110.