Splicing motivic zeta functions

Thomas Cauwbergs

doi:10.1007/s13163-016-0193-2

Revista Matemática Complutense

May 2016, Volume 29, Issue 2, pp 455–483 | Cite as

Splicing motivic zeta functions

Authors
Authors and affiliations

Thomas Cauwbergs

Article

First Online: 04 April 2016

148 Downloads
3 Citations

Abstract

The first part of the paper discusses how to lift the splicing formula of Némethi and Veys to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form, we prove a splicing formula for them, which specializes to this formula of Némethi and Veys. We also show that we cannot introduce a monodromic motivic zeta functions in terms of a (splice) diagram since it does not contain all the necessary information. In the last part we discuss the generalized monodromy conjecture of Némethi and Veys. The statement also holds for motivic zeta functions but it turns out that the analogous statement for monodromic motivic zeta functions is not correct. We illustrate this with some examples.

Keywords

Splicing Motivic zeta functions Monodromy conjecture

Mathematics Subject Classification

Primary 14E18 Secondary 14H50 32S05

This is a preview of subscription content, to check access.

Notes

Acknowledgments

I am very grateful to Wim Veys and Johannes Nicaise for their valuable suggestions. I am also very grateful for the suggestions made by the referees.

Appendix: Existence of the Picard morphism

It turns out that the class of a smooth and proper algebraic variety in the Grothendieck ring of varieties determines the class of its Picard scheme. In this Appendix we will prove this statement using the argument described by Ekedahl [14]. We do this since [14] was never published and to clarify several steps in his argument.

Let X be a smooth and proper algebraic variety over $\mathbb {C}$. Recall that the Picard functor ${{\mathrm{Pic}}}_{X\slash \mathbb {C}}$ is a functor from the category of locally Noetherian $\mathbb {C}$-schemes to the category of abelian groups defined by the formula

$$\begin{aligned} {{\mathrm{Pic}}}_{X\slash \mathbb {C}}(T) := {{\mathrm{Pic}}}(X_T) \slash {{\mathrm{Pic}}}(T) \end{aligned}$$

where $X_T = X \times _\mathbb {C}T$. It turns out that the associated sheaf is representable by an abelian scheme whose identity component is an abelian variety and whose group of geometric components is finitely generated. This scheme will be denoted by ${{\mathrm{Pic}}}(X)$. See [15, Part 5] for more information on the Picard scheme.

Definition

Define $ \mathcal {A}_{\mathbb {C}}$ as the abelian group generated by isomorphism classes of commutative algebraic group schemes over $\mathbb {C}$ whose identity component is an abelian variety and whose group of geometric components is finitely generated, subject to the relations $ [A \oplus B ] = [ A] + [B] $, where A and B are such commutative algebraic group schemes over $\mathbb {C}$ and $A \oplus B $ is the direct product of A and B.

This leads us to the main result.

Theorem A.1

There is a (unique) group homomorphism ${{\mathrm{Pic}}}:K_0({{\mathrm{Var}}}_{{\mathbb {C}}}) \rightarrow \mathcal {A}_{\mathbb {C}}$ such that ${{\mathrm{Pic}}}( [X]) = [{{\mathrm{Pic}}}(X)]$ for every smooth proper variety X, and it extends to a morphism ${{\mathrm{Pic}}}:\hat{\mathcal {M}}_{\mathbb {C}} \rightarrow \mathcal {A}_{\mathbb {C}}$.

This theorem provides us with a new technique to compare elements in the Grothendieck ring. Using this theorem, we find the existence of $\widetilde{{{\mathrm{Pic}}}} :\hat{\mathcal {M}}_{\mathbb {C}} \rightarrow {A}_{\mathbb {C}} $ which sends a smooth complete variety to the class of its jacobian. It is obtained by composing ${{\mathrm{Pic}}}$ and the morphism $\mathcal {A}_{\mathbb {C}} \rightarrow {A}_{\mathbb {C}} $, where this last map is defined by sending the class of a commutative algebraic group scheme (whose identity component is an abelian variety) to the class of his identity component.

The keystone of the proof is Bittner’s presentation of $K_0({{\mathrm{Var}}}_{{\mathbb {C}}}) $, which we restate here.

Theorem

[4, Theorem 3.1] The Grothendieck group of $\mathbb {C}$-varieties $K_0({{\mathrm{Var}}}_{{\mathbb {C}}})$ is isomorphic to the abelian group generated by the isomorphism classes of smooth projective $\mathbb {C}$-varieties subject to the relations $ [\emptyset ] = 0 $ and $ [{{\mathrm{Bl}}}_YX] - [E] = [X] - [Y]$, where X is smooth and projective, $Y\subset X$ is a closed smooth subvariety, ${{\mathrm{Bl}}}_Y X$ is the blow-up of X along Y and E is the exceptional divisor of this blow-up.

This implies the following presentation of $\mathcal {M}_{\mathbb {C}} = K_0({{\mathrm{Var}}}_{{\mathbb {C}}})[\mathbb {L}^{-1}]$: $\mathcal {M}_{\mathbb {C}}$ is generated as a group by the elements $ [X]\mathbb {L}^{-n}$ where X is smooth and projective and $n\in {\mathbb {Z}}_{>0}$, subject to the relations

$$\begin{aligned} \frac{ [X\times \mathbb {P}^{1}_{\mathbb {C}}] }{\mathbb {L}^{n+1} } = \frac{ [X] }{\mathbb {L}^{n} } + \frac{ [X ]}{\mathbb {L}^{n+1} }, \end{aligned}$$

where X is smooth and projective and $n\in {\mathbb {Z}}_{>0}$, and the relations

$$\begin{aligned} \frac{ [{{\mathrm{Bl}}}_YX]}{\mathbb {L}^n} - \frac{ [E]}{\mathbb {L}^n} = \frac{ [X]}{\mathbb {L}^n} - \frac{ [Y]}{\mathbb {L}^n}, \end{aligned}$$

where X, Y and E are as in the theorem and $n\in {\mathbb {Z}}_{>0}$.

Proof of Theorem A.1

We first show that the morphism defined by ${{\mathrm{Pic}}}( [X]) = [{{\mathrm{Pic}}}(X)]$ for X smooth and proper is well-defined as a map from $K_0({{\mathrm{Var}}}_{{\mathbb {C}}})$ to $ \mathcal {A}_{\mathbb {C}}$. Using the Bittner representation, we need to show that

$$\begin{aligned} {{\mathrm{Pic}}}({{\mathrm{Bl}}}_YX) \oplus {{\mathrm{Pic}}}(Y) = {{\mathrm{Pic}}}(X) \oplus {{\mathrm{Pic}}}(E) \end{aligned}$$

where X, Y and E are as in the theorem. Hence we need to show this on the level of associated fppf sheaves. But a blow-up is preserved under base change by a flat morphism and thus ${{\mathrm{Pic}}}({{\mathrm{Bl}}}_{Y_T} X_T) = {{\mathrm{Pic}}}(X_T ) \oplus {\mathbb {Z}}$ and ${{\mathrm{Pic}}}(E_T) = {{\mathrm{Pic}}}( Y_T ) \oplus {\mathbb {Z}}$ for any flat morphism $T \rightarrow \mathbb {C}$ which induces the wanted isomorphism [17, Exercises 7.9 and 8.5 on pp. 170 and 188].

The next step is to define ${{\mathrm{Pic}}}' :\mathcal {M}_{\mathbb {C}} \rightarrow \mathcal {A}_{\mathbb {C}}$ and prove that this is actually an extension of ${{\mathrm{Pic}}}$. Define ${{\mathrm{Pic}}}'( [X] \mathbb {L}^{-n} )$ as

$$\begin{aligned} A_{X}^{0,n} \oplus A_{X}^{c,n} \end{aligned}$$

for a projective and smooth variety X and $n \in {\mathbb {Z}}_{>0}$, where $A_{X}^{c,n}$ is the inverse image of the classes of type $(n+1, n+1)$ under the map $H^{2n+2}(X,{\mathbb {Z}}) \rightarrow H^{2n+2}(X,\mathbb {C})$ and $A_{X}^{0,n}$ is the Weil intermediate Jacobian associated to the Hodge structure on $H^{2n+1}(X,{\mathbb {Z}})$. See [31] for more information. As discussed, we need to verify the two types of relations. Consider X to be a projective smooth $\mathbb {C}$-variety.

First we verify the blow-up relations. This is a consequence of [30, Theorem 7.31] and its proof since it induces that

$$\begin{aligned} H^{k}(X,{\mathbb {Z}}) \oplus \left( \bigoplus _{i=0}^{r-2} H^{k-2i - 2}(Y,{\mathbb {Z}}) \right) \oplus H^{k}(Y,{\mathbb {Z}}) \end{aligned}$$

is both isomorphic to $H^{k}({{\mathrm{Bl}}}_Y X,{\mathbb {Z}}) \oplus H^{k}(Y,{\mathbb {Z}})$ and $H^{k}(X ,{\mathbb {Z}}) \oplus H^{k}(E,{\mathbb {Z}})$ as Hodge structures, where r is the codimension of Y in X.

Remark that the cup-products map [30, Theorem 11.38] for X and $ \mathbb {P}^{1}_{\mathbb {C}}$ induces an isomorphism

$$\begin{aligned} H^{i+2}(X \times \mathbb {P}^{1}_{\mathbb {C}} ,{\mathbb {Z}}) \cong H^{i+2}(X,{\mathbb {Z}}) \oplus H^{i}(X,{\mathbb {Z}}) \end{aligned}$$

as Hodge structures [12, p. 32].

Since the relations hold on the level of Hodge structures, they also hold for ${{\mathrm{Pic}}}'$ and thus ${{\mathrm{Pic}}}'$ is well-defined.

We show now that ${{\mathrm{Pic}}}'$ is indeed an extension of ${{\mathrm{Pic}}}$ and thus we will show that ${{\mathrm{Pic}}}(X) \cong A_{X}^{0,0} \oplus A_{X}^{c,0}$. Remark that the connected component ${{\mathrm{Pic}}}^0(X)$ is the classical Jacobian of X and thus isomorphic to $A_{X}^{0,0}$.

The Néron-Severi group ${{\mathrm{NS}}}(X)$ is defined by the short exact sequence

$$\begin{aligned} 0 \rightarrow {{\mathrm{Pic}}}^0(X) \rightarrow {{\mathrm{Pic}}}(X) \rightarrow {{\mathrm{NS}}}(X) \rightarrow 0 \end{aligned}$$

and is the group of components of ${{\mathrm{Pic}}}(X)$. This group can also be identified with the image of $ d :H^{1}(X,\mathcal {O}_X^\times ) \rightarrow H^{2}(X,{\mathbb {Z}})$. Now Lefschetz’s theorem on (1, 1)-classes [16, p. 163] implies that ${{\mathrm{NS}}}(X)$ is exactly $A_{X}^{c,0}$.

Since we are working over an algebraically closed field and ${{\mathrm{NS}}}(X)$ is a discrete finitely generated abelian group, the short exact sequence splits and thus ${{\mathrm{Pic}}}(X) \cong {{\mathrm{Pic}}}^0(X) \oplus {{\mathrm{NS}}}(X)$.

To conclude we remark that $H^{n}(X,{\mathbb {Z}}) = 0$ if $n > 2 \hbox {dim}\,X$ and thus ${{\mathrm{Pic}}}'( [X] \mathbb {L}^{-n}) = 0$ if $\dim (X) -n \le 0 $. This implies that ${{\mathrm{Pic}}}'$ sends every element of $F^0$ to 0 and thus ${{\mathrm{Pic}}}' $ can be extended to $\hat{\mathcal {M}}_{\mathbb {C}}$. $\square $

One of the results Ekedahl obtains with this is the fact that $\hat{\mathcal {M}}_{\mathbb {C}}$ is not a domain.

References

1.
Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J.: Torification and factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572 (2002). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
2.
Artal Bartolo, E., Cassou-Noguès, P., Luengo, I., Melle Hernández, A.: Monodromy conjecture for some surface singularities. Ann. Sci. École Norm. Sup. (4) 35(4), 605–640 (2002)MathSciNetzbMATHGoogle Scholar
3.
Artal Bartolo, E., Cassou-Noguès, P., Luengo, I., Melle Hernández, A.: Quasi-ordinary power series and their zeta functions. Mem. Am. Math. Soc 178(841), vi+85 (2005)MathSciNetzbMATHGoogle Scholar
4.
Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(4), 1011–1032 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
5.
Bories, B.: Zeta functions, Bernstein–Sato polynomials, and the monodromy conjecture. Ph.D. thesis, KU Leuven, Leuven (2013)Google Scholar
6.
Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties. In: Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)Google Scholar
7.
de Jong, J., Noot, R.: Jacobians with complex multiplication. In: Arithmetic Algebraic Geometry (Texel, 1989). Progr. Math., vol. 89, pp. 177–192. Birkhäuser, Boston (1991)Google Scholar
8.
Denef, J., Loeser, F., d’Euler-Poincaré, C.: fonctions zêta locales et modifications analytiques. J. Am. Math. Soc. 5(4), 705–720 (1992)Google Scholar
9.
Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7(3), 505–537 (1998)MathSciNetzbMATHGoogle Scholar
10.
Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. In: European Congress of Mathematics, Vol. I (Barcelona, 2000). Progr. Math., vol. 201, pp. 327–348. Birkhäuser, Basel (2001)Google Scholar
11.
Denef, J., Loeser, F.: Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
12.
Dimca, A.: Sheaves in topology. In: Universitext. Springer, Berlin (2004)Google Scholar
13.
Eisenbud, D., Neumann, W.: Three-dimensional link theory and invariants of plane curve singularities. In: Annals of Mathematics Studies, vol. 110. Princeton University Press, Princeton (1985)Google Scholar
14.
Ekedahl, T.: The Grothendieck group of algebraic stacks. 21 (2009). arXiv:0903.3143
15.
Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S.L., Nitsure, N., Vistoli, A.: Fundamental algebraic geometry: Grothendieck’s FGA explained, vol. 123. American Mathematical Society (AMS), Providence (2005). (English)Google Scholar
16.
Griffiths, P., Harris, J.: Principles of algebraic geometry. In: Wiley Classics Library. Wiley, New York (1994). (Reprint of the 1978 original)Google Scholar
17.
Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
18.
Loeser, F.: Fonctions d’Igusa $p$-adiques et polynômes de Bernstein. Am. J. Math. 110(1), 1–21 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
19.
Loeser, F.: Seattle lectures on motivic integration. In: Algebraic Geometry–Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, pp. 745–784. Am. Math. Soc., Providence (2009)Google Scholar
20.
Loeser, F.: Microlocal geometry and valued fields. Publ. Res. Inst. Math. Sci. 47(2), 613–627 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
21.
Mumford, D.: Abelian varieties. In: Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi (2008). [With appendices by C. P. Ramanujam and Yuri Manin, corrected reprint of the second (1974) edition]Google Scholar
22.
Némethi, A., Veys, W.: Generalized monodromy conjecture in dimension two. Geom. Topol. 16(1), 155–217 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
23.
Nicaise, J.: An introduction to $p$-adic and motivic zeta functions and the monodromy conjecture. In: Algebraic and Analytic Aspects of Zeta Functions and $L$-Functions, MSJ Mem., vol. 21, pp. 141–166. Math. Soc. Japan (2010)Google Scholar
24.
Rodrigues, B., Veys, W.: Poles of zeta functions on normal surfaces. Proc. Lond. Math. Soc. (3) 87(1), 164–196 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
25.
Schrauwen, R.: Series of singularities and their topology. Ph.D. thesis, Rijksuniversiteit Utrecht, Utrecht (1990)Google Scholar
26.
Schrauwen, R., Steenbrink, J., Stevens, J.: Spectral pairs and the topology of curve singularities. In: Complex Geometry and Lie Theory (Sundance, UT, 1989). Proc. Sympos. Pure Math., vol. 53, pp. 305–328. Am. Math. Soc., Providence (1991)Google Scholar
27.
Veys, W.: Zeta functions for curves and log canonical models. Proc. Lond. Math. Soc. (3) 74(2), 360–378 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
28.
Veys, W.: Zeta functions and “Kontsevich invariants” on singular varieties. Can. J. Math. 53(4), 834–865 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
29.
Veys, W.: Monodromy eigenvalues and zeta functions with differential forms. Adv. Math. 213(1), 341–357 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
30.
Voisin, C.: Hodge theory and complex algebraic geometry. I, English ed. In: Cambridge Studies in Advanced Mathematics, vol. 76. Cambridge University Press, Cambridge (2007). (Translated from the French by Leila Schneps)Google Scholar
31.
Weil, A.: On Picard varieties. Am. J. Math. 74, 865–894 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
32.
Włodarczyk, J.: Toroidal varieties and the weak factorization theorem. Invent. Math. 154(2), 223–331 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

Authors and Affiliations

Thomas Cauwbergs
- 1
Email author

1.Dept. WiskundeKU LeuvenLouvainBelgium

Splicing motivic zeta functions

Abstract

Keywords

Mathematics Subject Classification

Notes

Acknowledgments

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article