Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 017, 23 pages      arXiv:1207.6938      https://doi.org/10.3842/SIGMA.2016.017

Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three

Anda Degeratu a and Thomas Walpuski b
a) University of Freiburg, Mathematics Institute, Germany
b) Massachusetts Institute of Technology, Department of Mathematics, USA

Received June 02, 2015, in final form February 06, 2016; Published online February 15, 2016

Abstract
For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the crepant resolution using the Atiyah-Patodi-Singer index theorem. These results are dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288 (1990), 263-307] in dimension two.

Key words: crepant resolutions; HYM connections.

pdf (526 kb)   tex (36 kb)

References

1. Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69.
2. Bando S., Einstein-Hermitian metrics on noncompact Kähler manifolds, in Einstein Metrics and Yang-Mills Connections (Sanda, 1990), Lecture Notes in Pure and Appl. Math., Vol. 145, Dekker, New York, 1993, 27-33.
3. Bando S., Kasue A., Nakajima H., On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), 313-349.
4. Bartnik R., The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661-693.
5. Baum P., Fulton W., MacPherson R., Riemann-Roch and topological $K$ theory for singular varieties, Acta Math. 143 (1979), 155-192.
6. Blichfeldt H.F., Finite collineation groups: with an introduction to the theory of groups of operators and substitution groups, University of Chicago Press, Chicago, 1917.
7. Bridgeland T., King A., Reid M., The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), 535-554.
8. Bühler T., An introduction to the derived category, Notes to a series of lectures given at the Mirror Symmetry Learning Seminar, ETH Zürich, 2007, available at http://xwww.uni-math.gwdg.de/theo/intro-derived.pdf.
9. Calderbank D.M.J., Gauduchon P., Herzlich M., Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal. 173 (2000), 214-255, math.DG/9909116.
10. Craw A., Ishii A., Flops of $G$-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124 (2004), 259-307, math/0211360.
11. Degeratu A., Eta-invariants and Molien series for unimodular groups, Ph.D. Thesis, Massachusetts Institute of Technology, 2001.
12. Degeratu A., Mazzeo R., Fredholm theory for elliptic operators on quasi-asymptotically conical spaces, arXiv:1406.3465.
13. Gocho T., Nakajima H., Einstein-Hermitian connections on hyper-Kähler quotients, J. Math. Soc. Japan 44 (1992), 43-51.
14. Gonzalez-Sprinberg G., Verdier J.-L., Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983), 409-449.
15. Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York - Heidelberg, 1977.
16. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
17. Huybrechts D., Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006.
18. Ito Y., Nakajima H., McKay correspondence and Hilbert schemes in dimension three, Topology 39 (2000), 1155-1191, math.AG/9803120.
19. Joyce D.D., Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
20. King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515-530.
21. Kronheimer P.B., The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665-683.
22. Kronheimer P.B., Nakajima H., Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), 263-307.
23. McKay J., Graphs, singularities, and finite groups, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., Vol. 37, Amer. Math. Soc., Providence, R.I., 1980, 183-186.
24. Neeman A., Algebraic and analytic geometry, London Mathematical Society Lecture Note Series, Vol. 345, Cambridge University Press, Cambridge, 2007.
25. Sardo Infirri A.V., Partial resolutions of orbifold singularities via moduli spaces of HYM-type bundles, alg-geom/9610004.
26. Thomas R.P., Derived categories for the working mathematician, in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math., Vol. 23, Amer. Math. Soc., Providence, RI, 2001, 349-361, math.AG/0001045.
27. Toen B., $K$-théorie et cohomologie des champs algébriques, Ph.D. Thesis, Université Paul Sabatier, Toulouse, 1999.
28. Walpuski T., Gauge theory on ${\rm G}_2$-manifolds, Ph.D. Thesis, Imperial College London, 2013.
29. Yau S.S.-T., Yu Y., Gorenstein quotient singularities in dimension three, Mem. Amer. Math. Soc. 105 (1993), viii+88 pages.