## Duke Mathematical Journal

### Birational geometry and deformations of nilpotent orbits

Yoshinori Namikawa

#### Abstract

This is a continuation of [N2], where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution.

The purpose of this article is to describe the remainder. We first construct a deformation of the nilpotent orbit closure in a canonical manner, according to Brieskorn and Slodowy (see [S]), and next describe all its crepant simultaneous resolutions. This construction enables us to divide the whole space into a finite number of chambers.

Moreover, by using this construction, one can generalize the main result of [N2] to arbitrary Richardson orbits whose Springer maps have degree greater than $1$. New Mukai flops, different from those of types ${A}$, ${D}$, and ${E}_6$, appear in the birational geometry for such orbits

#### Article information

Source
Duke Math. J. Volume 143, Number 2 (2008), 375-405.

Dates
First available in Project Euclid: 26 May 2008

https://projecteuclid.org/euclid.dmj/1211819166

Digital Object Identifier
doi:10.1215/00127094-2008-022

Mathematical Reviews number (MathSciNet)
MR2420511

Zentralblatt MATH identifier
1140.14004

#### Citation

Namikawa, Yoshinori. Birational geometry and deformations of nilpotent orbits. Duke Math. J. 143 (2008), no. 2, 375--405. doi:10.1215/00127094-2008-022. https://projecteuclid.org/euclid.dmj/1211819166

#### References

• A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
• W. Borho and H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment Math. Helv. 54 (1979), 61--104.
• D. H. Collingwood and W. M. Mcgovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Math. Ser., Van Nostrand Reinhold, New York, 1993.
• B. Fu, Extremal contractions, stratified Mukai flops and Springer maps, Adv. Math. 213 (2007), 165--182.
• A. Fujiki, A generalization of an example of Nagata'' in Proceedings of the Algebraic Geometry Symposium (Fukuoka, Japan, 2000), Kyushu Univ., Fukuoka, Japan, 2000, 111--118.
• W. Hesselink, Polarizations in the classical groups, Math. Z. 160 (1978), 217--234.
• R. B. Howlett, Normalizers of parabolic subgroups of reflection groups, J. London Math. Soc. 21 (1980), 62--80.
• R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalized Hecke rings, Invent. Math. 58 (1980), 37--64.
• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972.
• Y. Kawamata, Crepant blowing-up of, $3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), 93--163.
• J. KolláR, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
• J. KolláR and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
• B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327--404.
• Y. Namikawa, Deformation theory of singular symplectic n-folds, Math. Ann. 319 (2001), 597--623.
• —, Birational geometry of symplectic resolutions of nilpotent orbits'' in Moduli Spaces and Arithmetic Geometry (Kyoto, 2004), Adv. Stud. Pure Math. 45, Math. Soc. Japan, Tokyo, 2006, 75--116.
• R. W. Richardson, Conjugacy classes of involutions in Coxeter groups, Bull. Austral. Math. Soc. 26 (1982), 1--15.
• P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Math. 815, Springer, Berlin, 1980.
• T. A. Springer and R. Steinberg, Conjugacy classes'' in Seminar on Algebraic Groups and Related Finite Groups (Princeton, N.J., 1968/69), Lecture Notes in Math. 131, Springer, Berlin, 1970, 167--266.