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Rationality of descendent series for Hilbert and Quot schemes of surfaces

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Abstract

Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series of virtual Euler characteristics was conjectured to be a rational function in Oprea and Pandharipande (in, Geom Topol. http://arxiv.org/abs/1903.08787) when X is simply connected. We conjecture here the rationality of more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. We prove the rationality of descendent series in Hilbert scheme cases for all curve classes and in Quot scheme cases when the curve class is 0.

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Notes

  1. See [20, 21, 35] for further developments.

  2. The conjecture can also be made for surfaces which are not simply connected, but we will not study non simply connected surfaces here (except in the \(\beta = 0 \) case).

  3. Property (ii) is proven in [25] for simply connected minimal surfaces of general type with \(p_g>0\) and a nonsingular canonical divisor. The assumptions other than \(p_g>0\) were removed in [18]. A similar analysis was done in [18] at the level of \(\chi _{-y}\)-genera.

  4. Property (iii) is proven in [26] for simply connected minimal elliptic surfaces. These assumptions were removed in [18] at the level of \(\chi _{-y}\)-genera.

  5. The simplest geometry \(X={\mathbb {P}}^2\) places numerical restrictions leading, at least a priori, to less precise results regarding the denominators of the answers.

  6. In the absence of (c), we have less control on the denominators of the rational functions thus obtained.

  7. As a consequence, the denominators of the series of Euler characteristics (3) are products of \(1-q\) and \(1-2q\) with various exponents. The same assertion holds true for the descendent series of Theorem 3. The example of Sect. 4.2.4 with \(\beta =0\) also has the same denominators.

  8. We have

    $$\begin{aligned} 0\rightarrow {\mathcal {T}}{\mathcal {o}}{\mathcal {r}}^{1}_{{\mathcal {C}}\,} (\Omega _{{\mathcal {C}}/{\mathcal {B}}}, {\mathcal {O}}_{{\mathcal {C}}_b}) \rightarrow {\mathcal {N}}\big |_{{\mathcal {C}}_b}\rightarrow \Omega _{X}\big |_{{\mathcal {C}}_b}\rightarrow \Omega _{{\mathcal {C}}_b}\rightarrow 0. \end{aligned}$$

    \({\mathcal {T}}or^1\) is supported on the finitely many singularities of \({\mathcal {C}}_b\). Since \({\mathcal {N}}\big |_{{\mathcal {C}}_b}\) is locally free, \({\mathcal {T}}or^1\) vanishes. Therefore, \({\mathcal {N}}\big |_{{\mathcal {C}}_b}\) is the conormal bundle.

  9. There are other roots which we will deal with later. See Eq. (29).

  10. The overall q shift does not affect rationality.

  11. This can also be seen via (5) since \(\text {Hilb}_{\beta }\) has negative virtual dimension.

  12. We can show

    $$\begin{aligned} {\mathsf {a}}=1-e_2t^2+2e_3t^3-3e_4t^4+\ldots \end{aligned}$$

    where \(e_i\) are the elementary symmetric functions in \(1, x_1, \ldots , x_{\ell }\). We do not explain the latter formula for \({\mathsf {a}}\) since it will not be used here.

  13. The same calculation can also be carried out using Theorem 2.

References

  1. Altman, A., Kleiman, S.: Compactifying the Picard scheme. Adv. Math. 35, 50–112 (1980)

    Article  MathSciNet  Google Scholar 

  2. Arbesfeld, N., Johnson, D., Lim, W., Oprea, D., Pandharipande, R.: The virtual \(K\)-theory of Quot schemes of surfaces. J. Geom. Phys. (2021). https://doi.org/10.1016/j.geomphys.2021.104154

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlsson, E.: Vertex operators and quasimodularity of Chern numbers on the Hilbert scheme. Adv. Math. 229, 2888–2907 (2012)

    Article  MathSciNet  Google Scholar 

  4. di Rocco, S.: k-very ample line bundles on del Pezzo surfaces. Math. Nach. 179, 47–56 (1996)

    Article  MathSciNet  Google Scholar 

  5. Duerr, M., Kabanov, A., Okonek, Ch.: Poincare invariants. Topology 46, 225–294 (2007)

    Article  MathSciNet  Google Scholar 

  6. Ellingsrud, G., Göttsche, L., Lehn, M.: On the cobordism class of the Hilbert scheme of a surface. J. Algorithm Geom. 10, 81–100 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math. 10, 511–521 (1968)

    Article  MathSciNet  Google Scholar 

  8. Fantechi, B., Göttsche, L.: Riemann–Roch theorems and elliptic genus for virtually smooth schemes. Geom. Topol. 14, 83–115 (2010)

    Article  MathSciNet  Google Scholar 

  9. Friedman, R.: Algebraic Sufaces and Holomorphic Vector Bundles. Springer, New York (1998)

    Book  Google Scholar 

  10. Gessel, I.: A combinatorial proof of the multivariable Lagrange inversion formula. J. Comb. Theory Ser. A 45, 178–195 (1987)

    Article  MathSciNet  Google Scholar 

  11. Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286, 193–207 (1990)

    Article  MathSciNet  Google Scholar 

  12. Göttsche, L., Kool, M.: Virtual refinements of the Vafa–Witten formula. Commun. Math. Phys. 376, 1–49 (2020)

    Article  MathSciNet  Google Scholar 

  13. Göttsche, L., Kool, M.: Refined SU(3) Vafa–Witten invariants and modularity. Pure Appl. Math. Q. 14, 467–513 (2018)

    Article  MathSciNet  Google Scholar 

  14. Göttsche, L., Shende, V.: Refined curve counting on complex surfaces. Geom. Topol. 18, 2245–2307 (2014)

    Article  MathSciNet  Google Scholar 

  15. Gronojnowski, I.: Instantons and affine algebras I. The Hilbert scheme and vertex operators. Math. Res. Lett. 3, 275–291 (1996)

    Article  MathSciNet  Google Scholar 

  16. Kool, M., Thomas, R.: Reduced classes and curve counting on surfaces I: Theory. Algorithm Geom. 1, 334–383 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Lim, W.: Virtual \(\chi _{-y}\)-genera of Quot schemes on surfaces. arXiv:2003.04429

  19. Marian, A., Oprea, D., Pandharipande, R.: Segre classes and Hilbert schemes of points. Ann. Sci. l’ENS 50, 239–267 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Marian, A., Oprea, D., Pandharipande, R.: The combinatorics of Lehn’s conjecture. J. Math. Soc. Jpn. 71, 299–308 (2019)

    Article  MathSciNet  Google Scholar 

  21. Marian, A., Oprea, D., Pandharipande, R.: Higher rank Segre integrals over the Hilbert scheme of points. J. Eur. Math. Soc. (to appear). arXiv:1712.02382

  22. Moreira, M., Oblomkov, A., Okounkov, A., Pandharipande, R.: Virasoro constraints for stable pairs on toric 3-folds. arXiv:2008.12514

  23. Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. Math. 145, 379–388 (1997)

    Article  MathSciNet  Google Scholar 

  24. Oblomkov, A., Okounkov, A., Pandharipande, R.: GW/PT descendent correspondence via vertex operators. Commun. Math. Phys. 374, 1321–1359 (2020)

    Article  MathSciNet  Google Scholar 

  25. Oprea, D., Pandharipande, R.: Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics. Geom. Topol. (to appear). arXiv:1903.08787

  26. Oprea, D., Pandharipande, R.: Private Conversation. ETH Zurich, Zurich (2019)

    Google Scholar 

  27. Pandharipande, R.: Descendents for stable pairs on 3-folds, modern geometry: a celebration of the work of Simon Donaldson. Proc. Sym. Pure Math. 99, 251–288 (2018)

    MATH  Google Scholar 

  28. Pandharipande, R., Pixton, A.: Descendents for stable pairs on \(3\)-folds: rationality. Comput. Math. 149, 81–124 (2013)

    MATH  Google Scholar 

  29. Pandharipande, R., Pixton, A.: Descendent theory for stable pairs on toric \(3\)-folds. J. Math. Soc. Jpn. 65, 1337–1372 (2013)

    Article  MathSciNet  Google Scholar 

  30. Pandharipande, R., Pixton, A.: Gromov–Witten/Pairs descendent correspondence for toric \(3\)-folds. Geom. Topol. 18, 2747–2821 (2014)

    Article  MathSciNet  Google Scholar 

  31. Pandharipande, R., Thomas, R.: Curve counting via stable pairs in the derived category. Invent. Math. 178, 407–447 (2009)

    Article  MathSciNet  Google Scholar 

  32. Pandharipande, R., Thomas, R.: Stable pairs and BPS invariants. J. Am. Math. Soc. 23, 267–297 (2010)

    Article  MathSciNet  Google Scholar 

  33. Thomas, R.P.: A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations. J. Diff. Geom. 54, 367–438 (2000)

    MathSciNet  MATH  Google Scholar 

  34. Tanaka, Y., Thomas, R.P.: Vafa–Witten invariants for projective surfaces I: stable case. J. Algorithm Geom. 29, 603–668 (2020)

    Article  MathSciNet  Google Scholar 

  35. Voisin, C.: Segre classes of tautological bundles on Hilbert schemes of points of surfaces. Algorithm Geom. 6, 186–195 (2019)

    Article  Google Scholar 

  36. Vafa, C., Witten, E.: A strong coupling test of S-duality. Nucl. Phys. B 431, 3–77 (1994)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Our study of the virtual Euler characteristics of the Quot scheme of surfaces was motivated in part by the Euler characteristic calculations of L. Göttsche and M. Kool [12, 13] for the moduli spaces of rank 2 and 3 stable sheaves on surfaces. We thank A. Marian, W. Lim, A. Oblomkov, A. Okounkov, and R. Thomas for related discussions. D. J. was supported by SNF-200020-182181. D. O. was supported by the NSF through Grant DMS 1802228. R.P. was supported by the Swiss National Science Foundation and the European Research Council through Grants SNF-200020-182181, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. We thank the Shanghai Center for Mathematical Science at Fudan University for a very productive visit in September 2018 at the start of the project. The project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 Research and Innovation Program (Grant No. 786580).

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Zurich, Switzerland

    Drew Johnson & Rahul Pandharipande

  2. Department of Mathematics, University of California, San Diego, USA

    Dragos Oprea

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  1. Drew Johnson
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Correspondence to Dragos Oprea.

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Johnson, D., Oprea, D. & Pandharipande, R. Rationality of descendent series for Hilbert and Quot schemes of surfaces. Sel. Math. New Ser. 27, 23 (2021). https://doi.org/10.1007/s00029-021-00638-1

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