A note on functional equations for zeta functions with values in Chow motives

Heinloth, Franziska

doi:10.5802/aif.2318

Cited by 49 publications

(42 citation statements)

References 13 publications

(16 reference statements)

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“…For X a point pt, properties (i) and (iii) endow the Grothendieck group (with respect to direct sums)K 0 (A( pt)) with a pre-lambda ring structure defined by (compare [26]):…”

Section: Is a Complex Algebraic Variety Or A Compact Complex Analyticmentioning

confidence: 99%

Twisted genera of symmetric products

Maxim

Schürmann

2011

Sel. Math. New Ser.

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We give a new proof of formulae for the generating series of (Hodge) genera of symmetric products X (n) with coefficients, which hold for complex quasiprojective varieties X with any kind of singularities and which include many of the classical results in the literature as special cases. Important specializations of our results include generating series for extensions of Hodge numbers and Hirzebruch's χ y -genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. Our proof applies to more general situations and is based on equivariant Künneth formulae and pre-lambda structures on the coefficient theory of a point,K 0 (A( pt)), with A( pt) a Karoubian Q-linear tensor category. Moreover, Atiyah's approach to power operations in K -theory also applies in this context toK 0 (A( pt)), giving a nice description of the important related Adams operations. This last approach also allows us to introduce very interesting coefficients on the symmetric products X (n) .

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“…For X a point pt, properties (i) and (iii) endow the Grothendieck group (with respect to direct sums)K 0 (A( pt)) with a pre-lambda ring structure defined by (compare [26]):…”

Section: Is a Complex Algebraic Variety Or A Compact Complex Analyticmentioning

confidence: 99%

Twisted genera of symmetric products

Maxim

Schürmann

2011

Sel. Math. New Ser.

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“…These equations, obtained by M. Kapranov [3] and F. Heinloth [4], follow from the duality theory on curves and Abelian varieties. It should be noted that they have an origin different from our approach.…”

Section: §1 Introductionmentioning

confidence: 99%

Motivic integrals and functional equations

Gorsky

2008

St. Petersburg Math. J.

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Abstract. A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived by using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to multiplication by a constant, and there is a simple recursive algorithm to find its coefficients. The method is fairly universal and gives, for example, equations for the integral corresponding to the intersection number over the space of pairs of arcs and over the space of unordered collections of arcs. §1. Introduction Motivic integration, introduced by M. Kontsevich, is a powerful tool for exploring the space of formal arcs on a given variety. Motivic integrals provide the generating series for motivic measures of level sets of some arc invariants. There are examples of such integrals that can be calculated explicitly (see, e.g., [1]). In more general situations the values of such integrals are unknown; however, some of them satisfy functional equations if some auxiliary variables are introduced.In this paper, with the help of the Denef-Loeser formula for the change of variables (see [1]), a functional equation is derived for the motivic integral that gives the generating series corresponding to the Milnor number of a plane curve. Its solution is unique up to multiplication by a constant, and there is a simple algorithm to express the solution's coefficients via the initial data. For example, this means that the solution satisfies some partial differential equations. The equation obtained gives a method for computing the motivic measure (and consequently, the Hodge-Deligne polynomial) of the stratum {µ = const} in the space of plane curves. Some examples are considered in §4.A similar idea gives some other equations, for example, an equation for the integral corresponding to the intersection number over the space of pairs of arcs. Moreover, using the notion of the power structure on the Grothendieck ring (see [2]), we introduce a motivic measure on the space of unordered collections of arcs. We also derive a curious equation for the integral corresponding to the intersection number in this case.Some generating series with coefficients in the Grothendieck ring of varieties K 0 (V ar C ) (or in the Grothendieck ring of Chow motives) satisfy functional equations similar to the functional equation for the Hasse-Weil zeta function. These equations, obtained by M. Kapranov [3] and F. Heinloth [4], follow from the duality theory on curves and Abelian varieties. It should be noted that they have an origin different from our approach. §2. Motivic measure Let L = L C 2 ,0 be the space of arcs at the origin on the plane. It is the set of pairs (x(t), y(t)) of formal power series (without degree 0 terms).2000 Mathematics Subject Classification. Primary 32S45, 28B10.

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“…Кроме того, Хайнлот доказала, что для таких мотивов дзета-функция удовлетворяет функциональному уравнению (см. [4]). …”

Section: введение в [1] и [2]unclassified

“…То-гда в T определены внешние и симметрические степени. Поскольку T аддитивна, K s 0 (T ) имеет две канонические лямбда-структуры λ ± по лемме 4.1 из [4] (см. ниже).…”

Section: введение в [1] и [2]unclassified

О Дзета-Функциях В Триангулированных Категориях

Guletskiĭ¹

2010

Mat. Zametki

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Том 87 выпуск 3 март 2010 УДК 513.6 О дзета-функциях в триангулированных категориях В. И. Гулецкий В работе доказывается "два-из-трех" свойство рациональности мотивной дзе-та-функции в выделенных треугольниках в категории Воеводского DM . В качестве приложения мы получаем рациональность мотивной дзета-функ-ции для всех многообразий, мотивы которых лежат в насыщенной триангули-рованной тензорной подкатегории, порожденной мотивами квазипроективных кривых в DM . Вместе с результатом П. О'Сулливана это дает первый при-мер многообразия, мотив которого не является конечномерным, а его мотивная дзета-функция, тем не менее, рациональна.Библиография: 25 названий.

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A note on functional equations for zeta functions with values in Chow motives

Cited by 49 publications

References 13 publications

Twisted genera of symmetric products

Twisted genera of symmetric products

Motivic integrals and functional equations

О Дзета-Функциях В Триангулированных Категориях

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