scholarly journals Zeta Functions and ‘Kontsevich Invariants’ on Singular Varieties

2001 ◽  
Vol 53 (4) ◽  
pp. 834-865 ◽  
Author(s):  
Willem Veys
Keyword(s):  
Zeta Function ◽  
Regular Function ◽  
Zeta Functions ◽  
Singular Locus ◽  
Normal Surface ◽  
Grothendieck Ring ◽  
Singular Varieties ◽  
Hodge Numbers ◽  
Topological Zeta Function ◽  
Motivic Zeta Function

AbstractLet X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any ℚ-Gorenstein variety X we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective ℚ-Cartier divisors on X whose support contains the singular locus of X.

scholarly journals MOTIVIC HILBERT ZETA FUNCTIONS OF CURVES ARE RATIONAL

2018 ◽  
Vol 19 (3) ◽  
pp. 947-964
Author(s):  
Dori Bejleri ◽  
Dhruv Ranganathan ◽  
Ravi Vakil
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Zeta Functions ◽  
Hilbert Schemes ◽  
Grothendieck Ring

The motivic Hilbert zeta function of a variety $X$ is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points on $X$. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.

scholarly journals Computing motivic zeta functions on log smooth models

2019 ◽  
Vol 295 (1-2) ◽  
pp. 427-462 ◽  
Author(s):  
Emmanuel Bultot ◽  
Johannes Nicaise
Keyword(s):  
Zeta Function ◽  
Recent Work ◽  
Explicit Formula ◽  
Essential Role ◽  
Zeta Functions ◽  
Smooth Model ◽  
Motivic Zeta Functions ◽  
Motivic Zeta Function ◽  
Special Case

Abstract We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results.

scholarly journals ALGEBRO-GEOMETRIC FEYNMAN RULES

2011 ◽  
Vol 08 (01) ◽  
pp. 203-237 ◽  
Author(s):  
PAOLO ALUFFI ◽  
MATILDE MARCOLLI
Keyword(s):  
General Procedure ◽  
Zeta Functions ◽  
Feynman Rule ◽  
Characteristic Classes ◽  
Partition Functions ◽  
Grothendieck Ring ◽  
Projective Varieties ◽  
Singular Varieties ◽  
Feynman Rules ◽  
Graph Hypersurface

We give a general procedure to construct "algebro-geometric Feynman rules", that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining "motivic Feynman rules". We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.

scholarly journals Geometric determination of the Poles of Highest and second Highest order of Hodge And motivic Zeta functions

2004 ◽  
Vol 176 ◽  
pp. 1-18
Author(s):  
B. Rodrigues
Keyword(s):  
Zeta Function ◽  
Arbitrary Dimension ◽  
Zeta Functions ◽  
Motivic Zeta Functions ◽  
Hodge Polynomials ◽  
Motivic Zeta Function

AbstractTo any f ∈ ℂ[x1, … ,xn] \ ℂ with f(0) = 0 one can associate the motivic zeta function. Another interesting singularity invariant of f-1{0} is the zeta function on the level of Hodge polynomials, which is actually just a specialization of the motivic one. In this paper we generalize for the Hodge zeta function the result of Veys which provided for n = 2 a complete geometric determination of the poles. More precisely we give in arbitrary dimension a complete geometric determination of the poles of order n − 1 and n. We also show how to obtain the same results for the motivic zeta function.

Motivic zeta functions in additive monoidal categories

2011 ◽  
Vol 9 (3) ◽  
pp. 459-473 ◽  
Author(s):  
Kenichiro Kimura ◽  
Shun-ichi Kimura ◽  
Nobuyoshi Takahashi
Keyword(s):  
Zeta Function ◽  
Monoidal Category ◽  
Zeta Functions ◽  
Monoidal Categories ◽  
Symmetric Monoidal Category ◽  
Motivic Zeta Functions ◽  
Motivic Zeta Function ◽  
Coefficient Ring

AbstractLet C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C. We study the problem of rationality of the motivic zeta function ζx(t) of X. Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζx(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

scholarly journals A New Determinant Expression for the Weighted Bartholdi Zeta Function of a Digraph

10.37236/2649 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Iwao Sato ◽  
Hideo Mitsuhasi ◽  
Hideaki Morita
Keyword(s):  
Zeta Function ◽  
Zeta Functions

We consider the weighted Bartholdi zeta function of a digraph $D$, and give a new determinant expression of it. Furthermore, we treat a weighted $L$-function of $D$, and give a new determinant expression of it. As a corollary, we present determinant expressions for the Bartholdi edge zeta functions of a graph and a digraph.

scholarly journals Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1431
Author(s):  
Junesang Choi ◽  
Recep Şahin ◽  
Oğuz Yağcı ◽  
Dojin Kim
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Zeta Functions ◽  
Integral Representations ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
The One ◽  
Function Of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

scholarly journals Second variation of Selberg zeta functions and curvature asymptotics

2019 ◽  
Vol 57 (1) ◽  
pp. 23-60
Author(s):  
Ksenia Fedosova ◽  
Julie Rowlett ◽  
Genkai Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Vector Bundle ◽  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Second Variation ◽  
Selberg Zeta Function ◽  
The Second Variation

Abstract We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, Z(s), on Teichmüller space. We then use this formula to determine the asymptotic behavior as $$\mathfrak {R}s \rightarrow \infty $$Rs→∞ of the second variation. As a consequence, for $$m \in {\mathbb {N}}$$m∈N, we obtain the complete expansion in m of the curvature of the vector bundle $$H^0(X_t, {\mathcal {K}}_t)\rightarrow t\in {\mathcal {T}}$$H0(Xt,Kt)→t∈T of holomorphic m-differentials over the Teichmüller space $${\mathcal {T}}$$T, for m large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $$O(m^2 \mathrm{e}^{-l_0 m}),$$O(m2e-l0m), where $$l_0$$l0 is the length of the shortest closed hyperbolic geodesic.

Sign in / Sign up

Export Citation Format

Share Document