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Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles

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Abstract

We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.

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References

  1. Addington, N.: New derived symmetries of some hyperkähler varieties. Algebr. Geom. 3(2), 223–260 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc 14(3), 535–554 (2001). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  3. Briançon, J.: Description de \(H{\rm ilb}^{n}C\{x, y\}\). Invent. Math. 41(1), 45–89 (1977)

    Article  MathSciNet  Google Scholar 

  4. Chen, J.-C.: Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities. J. Differ. Geom. 61(2), 227–261 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cautis, S., Licata, A.: Heisenberg categorification and Hilbert schemes. Duke Math. J. 161(13), 2469–2547 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dalbec, J.: Multisymmetric functions. Beiträge Algebra Geom. 40(1), 27–51 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Danila, G.: Sections du fibré déterminant sur l’espace de modules des faisceaux semi-stables de rang 2 sur le plan projectif. Ann. Inst. Fourier (Grenoble) 50(5), 1323–1374 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Danila, G.: Sur la cohomologie d’un fibré tautologique sur le schéma de Hilbert d’une surface. J. Algebraic Geom. 10(2), 247–280 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Elagin, A.D.: Semi-orthogonal decompositions for derived categories of equivariant coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 73(5), 37–66 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Elagin, A.D.: On equivariant triangulated categories. arXiv:1403.7027, (2014)

  11. Fulton, W., Harris, J.: Representation theory, volume 129 of Graduate Texts in Mathematics. Springer, New York, (1991). A first course, Readings in Mathematics

  12. Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math 90, 511–521 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  13. Haiman, M.: Macdonald polynomials and geometry. In: New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97), volume 38 of Math. Sci. Res. Inst. Publ., pp. 207–254. Cambridge Univ. Press, Cambridge (1999)

  14. Haiman, Mark: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc 14(4), 941–1006 (2001). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006)

  16. Krug, A., Ploog, D., Sosna, P.: Derived categories of resolutions of cyclic quotient singularities. Q. J. Math. (2017). https://doi.org/10.1093/qmath/hax048/4675118

  17. Krug, A.: Extension groups of tautological sheaves on Hilbert schemes. J. Algebr. Geom. 23(3), 571–598 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Krug, A.: \({\mathbb{P}}\)-functor versions of the Nakajima operators. arXiv:1405.1006 (2014)

  19. Krug, A.: Tensor products of tautological bundles under the Bridgeland-King-Reid-Haiman equivalence. Geom. Dedicata 172, 245–291 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Krug, A.: On derived autoequivalences of Hilbert schemes and generalized Kummer varieties. Int. Math. Res. Not. IMRN 20, 10680–10701 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Krug, A.: Symmetric quotient stacks and Heisenberg actions. Math. Z. (2017). https://doi.org/10.1007/s00209-017-1874-3.

  22. Krug, A., Sosna, P.: On the derived category of the Hilbert scheme of points on an Enriques surface. Selecta Math. (N.S.) 21(4), 1339–1360 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kool, M., Shende, V., Thomas, R.P.: A short proof of the Göttsche conjecture. Geom. Topol. 15(1), 397–406 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lehn, M.: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136(1), 157–207 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lehn, M., Sorger, C.: Symmetric groups and the cup product on the cohomology of Hilbert schemes. Duke Math. J. 110(2), 345–357 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lehn, M., Sorger, C.: The cup product of Hilbert schemes for \(K3\) surfaces. Invent. Math. 152(2), 305–329 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meachan, C.: Derived autoequivalences of generalised Kummer varieties. Math. Res. Lett. 22(4), 1193–1221 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Markman, E., Mehrotra, S.: Integral transforms and deformations of K3 surfaces. arXiv:1507.03108 (2015)

  29. Marian, A., Oprea, D.: A tour of theta dualities on moduli spaces of sheaves. In: Curves and abelian varieties, volume 465 of Contemp. Math., pp. 175–201. Am. Math. Soc., Providence, RI (2008)

  30. Nagaraj, D.S.: Vector bundles on symmetric product of curves. arXiv:1702.05294 (2017)

  31. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces, volume 18 of University Lecture Series. American Mathematical Society, Providence, RI (1999)

  32. Ploog, D.: Equivariant autoequivalences for finite group actions. Adv. Math. 216(1), 62–74 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Reid, M.: La correspondance de McKay. Astérisque, vol. 1999/2000 (276), pp. 53–72. Séminaire Bourbaki (2002)

  34. Rennemo, J.V.: Universal polynomials for tautological integrals on Hilbert schemes. Geom. Topol. 21(1), 253–314 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Scala, L.: Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles. Duke Math. J. 150(2), 211–267 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Scala, L.: Some remarks on tautological sheaves on Hilbert schemes of points on a surface. Geom. Dedicata 139, 313–329 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Scala, L.: Higher symmetric powers of tautological bundles on Hilbert schemes of points on a surface. arXiv:1502.07595 (2015)

  38. Scala, L.: Notes on diagonals of the product and symmetric variety of a surface. arXiv:1510.04889 (2015)

  39. Schwarzenberger, R.L.E.: Vector bundles on the projective plane. Proc. Lond. Math. Soc. 3(11), 623–640 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schlickewei, U.: Stability of tautological vector bundles on Hilbert squares of surfaces. Rend. Semin. Mat. Univ. Padova 124, 127–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Stapleton, D.: Geometry and stability of tautological bundles on Hilbert schemes of points. Algebra Numb. Theory 10(6), 1173–1190 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wandel, M.: Stability of tautological bundles on the Hilbert scheme of two points on a surface. Nagoya Math. J. 214, 79–94 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wandel, M.: Tautological sheaves: Stability, moduli spaces and restrictions to generalised Kummer varieties. Osaka J. Math. 53(4), 889–910 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Wang, Z., Zhou, J.: Tautological sheaves on Hilbert schemes of points. J. Algebr. Geom. 23(4), 669–692 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thanks Jörg Schürmann for interesting discussions and Sönke Rollenske for comments on the text. He also thanks the referee for helpful comments.

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Correspondence to Andreas Krug.

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Communicated by Vasudevan Srinivas.

Appendix A. Computations with power series

Appendix A. Computations with power series

Given a power series F(Q), we denote by \(F(Q)_{\mid Q^n}\) the coefficient of \(Q^n\). The verification that the two formulae of Corollary 4.6 are equivalent comes down to the following

Proposition A.1

$$\begin{aligned}&(-1)^{k+\ell }\exp \left( \sum _{r=1}^\infty \chi (\Lambda _{-v^r} K, \Lambda _{-u^r} L) \frac{Q^r}{r} \right) _{\mid v^ku^\ell Q^n} \\&\qquad = \sum _{i=\max \{0,k+\ell -n\}}^{\min \{k,\ell \}}s^i\chi (K,L)\cdot \lambda ^{k-i}\chi (K^\vee )\cdot \lambda ^{\ell -i}\chi (L)\cdot s^{n+i-k-\ell }\chi (\mathcal O_X) \end{aligned}$$

For the proof, we use two simple auxiliary lemmas.

Lemma A.2

$$\begin{aligned} \exp \left( \sum _{r=1}^\infty \frac{1}{r} Q^r\right) =\frac{1}{1-Q}\,. \end{aligned}$$

Proof

One way to see this is to apply the logarithm to both sides. \(\square \)

Lemma A.3

For \(k\in \mathbb {N}\) and \(\chi \in \mathbb {C}\), we have

  1. (i)

    \(s^k\chi =(-1)^k \lambda ^k(-\chi )\) ,

  2. (ii)

    \((1+Q)^\chi _{\mid Q^k}=\lambda ^k\chi \)  ,

  3. (iii)

    \((1-Q)^{-\chi }_{\mid Q^k}=s^k\chi \) .

Proof

The verification of (i) is a direct computation using Definition 4.5 of the numbers \(s^k\chi \) and \(\lambda ^k\chi \). Part (ii) is the binomial coefficient theorem. Part (iii) follows from (i) and (ii). \(\square \)

Proof of Proposition A.1

We have

$$\begin{aligned} \chi (\Lambda _{-v^r} K, \Lambda _{-u^r} L) Q^r= & {} \chi (K,L)(vuQ)^r -\chi (K^\vee )(vQ)^r \\&-\chi (L)(uQ)^r+ \chi (\mathcal O_X)Q^r\,. \end{aligned}$$

Hence, by Lemma A.2, we get

$$\begin{aligned}&\exp \left( \sum _{r=1}^\infty \chi (\Lambda _{-v^r} K, \Lambda _{-u^r} L) \frac{Q^r}{r} \right) \\&\quad =(1-vuQ)^{-\chi (K,L)} (1-vQ)^{\chi (K^\vee )}(1-uQ)^{\chi (L)}(1-Q)^{-\chi (\mathcal O_X)}. \end{aligned}$$

Now, the assertion follows using Lemma A.3. \(\square \)

The verification of Remark 5.6 is very similar.

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Krug, A. Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles. Math. Ann. 371, 461–486 (2018). https://doi.org/10.1007/s00208-018-1660-5

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