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On singularity properties of convolutions of algebraic morphisms

  • Itay Glazer
  • Yotam I. HendelEmail author
Article

Abstract

Let K be a field of characteristic zero, X and Y be smooth K-varieties, and let V be a finite dimensional K-vector space. For two algebraic morphisms \(\varphi :X\rightarrow V\) and \(\psi :Y\rightarrow V\) we define a convolution operation, \(\varphi *\psi :X\times Y\rightarrow V\), by \(\varphi *\psi (x,y)=\varphi (x)+\psi (y)\). We then study the singularity properties of the resulting morphism, and show that as in the case of convolution in analysis, it has improved smoothness properties. Explicitly, we show that for any morphism \(\varphi :X\rightarrow V\) which is dominant when restricted to each irreducible component of X, there exists \(N\in \mathbb {N}\) such that for any \(n>N\) the nth convolution power \(\varphi ^{n}:=\varphi *\dots *\varphi \) is a flat morphism with reduced geometric fibers of rational singularities (this property is abbreviated (FRS)). By a theorem of Aizenbud and Avni, for \(K=\mathbb {Q}\), this is equivalent to good asymptotic behavior of the size of the \(\mathbb {Z}/p^{k}\mathbb {Z}\)-fibers of \(\varphi ^{n}\) when ranging over both p and k. More generally, we show that given a family of morphisms \(\{\varphi _{i}:X_{i}\rightarrow V\}\) of complexity \(D\in \mathbb {N}\) (i.e. that the number of variables and the degrees of the polynomials defining \(X_{i}\) and \(\varphi _{i}\) are bounded by D), there exists \(N(D)\in \mathbb {N}\) such that for any \(n>N(D)\), the morphism \(\varphi _{1}*\dots *\varphi _{n}\) is (FRS).

Mathematics Subject Classification

03C98 14B05 14E18 11G25 14G05 

Notes

Acknowledgements

We thank Moshe Kamenski and Raf Cluckers for enlightening conversations about the model theoretic settings. We thank Nir Avni for numerous helpful discussions, as well as for proposing this problem together with Rami Aizenbud. A large part of this work was carried out while visiting the mathematics department at Northwestern university, we thank them and Nir for their hospitality. Finally we wish to thank our teacher Rami Aizenbud for answering various questions and for helping to shape many of the ideas in this paper. We benefited from his guidance deeply. We also wish to thank the anonymous referees for their insightful comments and remarks, and in particular for suggesting the alternative proof of Theorem 5.2 (see Sect. 5.2). Both authors where partially supported by ISF Grant 687/13, BSF Grant 2012247 and a Minerva Foundation Grant.

References

  1. 1.
    Aizenbud, A., Avni, N.: Representation growth and rational singularities of the moduli space of local systems. Invent. Math. 204(1), 245–316 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aizenbud, A., Avni, N.: Counting points of schemes over finite rings and counting representations of arithmetic lattices. Duke Math. J. 167(14), 2721–2743 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ax, J.: The elementary theory of finite fields. Ann. Math. (2) 88, 239–271 (1968)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berman, M.N., Derakhshan, J., Onn, U., Paajanen, P.: Uniform cell decomposition with applications to Chevalley groups. J. Lond. Math. Soc. (2) 87(2), 586–606 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cluckers, R.: Presburger sets and \(p\)-minimal fields. J. Symb. Logic 68(1), 153–162 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cluckers, R.: Multivariate Igusa theory: decay rates of exponential sums. Int. Math. Res. Not. 76, 4093–4108 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cluckers, R., Gordon, J., Halupczok, I.: Integrability of oscillatory functions on local fields: transfer principles. Duke Math. J. 163(8), 1549–1600 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cluckers, R., Gordon, J., Halupczok, I.: Motivic functions, integrability, and applications to harmonic analysis on \(p\)-adic groups. Electron. Res. Announc. Math. Sci. 21, 137–152 (2014)MathSciNetGoogle Scholar
  9. 9.
    Cluckers, R., Gordon, J., Halupczok, I.: Transfer principles for bounds of motivic exponential functions. In Families of automorphic forms and the trace formula, Simons Symp., pp. 111–127. Springer, Cham (2016)Google Scholar
  10. 10.
    Cluckers, R., Gordon, J., Halupczok, I.: Uniform analysis on local fields and applications to orbital integrals. Trans. Am. Math. Soc. Ser. B 5, 125–166 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cluckers, R., Loeser, F.: Constructible motivic functions and motivic integration. Invent. Math. 173(1), 23–121 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cluckers, R., Loeser, F.: Constructible exponential functions, motivic Fourier transform and transfer principle. Ann. Math. (2) 171(2), 1011–1065 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Denef, J.: On the degree of Igusa’s local zeta function. Am. J. Math. 109(6), 991–1008 (1987)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Elkik, R.: Singularités rationnelles et déformations. Invent. Math. 47(2), 139–147 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Glazer, I.: On rational singularities and counting points of schemes over finite rings. Algebra Number Theory. arXiv:1502.07004 (to appear)
  16. 16.
    Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Etudes Sci. Publ. Math. 32, 361 (1967)zbMATHGoogle Scholar
  17. 17.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
  18. 18.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I. Ann. Math. (2) 79, 109–203 (1964)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. II. Ann. Math. (2) 79, 205–326 (1964)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Igusa, J.: An Introduction to the Theory of Local Zeta Functions, Volume 14 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, International Press, Providence, Cambridge (2000)Google Scholar
  21. 21.
    Kempf, G., Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)CrossRefGoogle Scholar
  22. 22.
    Maxim, L.., Saito, M., Schürmann, J.: Thom–Sebastiani theorems for filtered \({\cal{D}}\)-modules and for multiplier ideals. Int. Math. Res. Not. rny032 (2018).  https://doi.org/10.1093/imrn/rny032
  23. 23.
    Mustaţă, M.: Jet schemes of locally complete intersection canonical singularities. Invent. Math. 145(3), 397–424 (2001). With an appendix by David Eisenbud and Edward FrenkelMathSciNetCrossRefGoogle Scholar
  24. 24.
    Mustaţă, M.: IMPANGA lecture notes on log canonical thresholds. In Contributions to algebraic geometry, EMS Ser. Congr. Rep., pp. 407–442. European Mathematical Society, Zürich (2012). Notes by Tomasz SzembergGoogle Scholar
  25. 25.
    Pas, J.: Uniform \(p\)-adic cell decomposition and local zeta functions. J. Reine Angew. Math. 399, 137–172 (1989)MathSciNetzbMATHGoogle Scholar
  26. 26.
    The Stacks Project Authors.: Stack Project. https://stacks.math.columbia.edu (2018)

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

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