A proof of the integral identity conjecture, II

Lê, Quy Thuong

doi:10.1016/j.crma.2017.10.005

Cited by 3 publications

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“…His arguments in fact still work in our situation. Moreover, there are methods more direct to prove this lemma, such as combinatorics or Cluckers-Loeser's computations for the constructible motivic functions in [1,Section 4] together with the version with action in [14].…”

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Euler reflexion formulas for motivic multiple zeta functions

Thuong¹,

Duc

2017

J. Algebraic Geom.

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We introduce a new notion of -product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions. We also show that the -product is associative in the class of motivic multiple zeta functions.Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the -product and motivic multiple zeta functions, and it is proved for both univariate and multivariate cases by using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well known motivic Thom-Sebastiani theorem.

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confidence: 99%

Euler reflexion formulas for motivic multiple zeta functions

Thuong¹,

Duc

2017

J. Algebraic Geom.

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“…Recently, by using Cluckers-Loeser's motivic integration [5] we have provided a new proof of the integral identity conjecture (in the framework of Mμ k,loc ) for any field k of characteristic zero (cf. [21]). …”

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A Short Survey on the Integral Identity Conjecture and Theories of Motivic Integration

Thuong

2016

Acta Math Vietnam

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Abstract. In Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau varieties, the integral identity conjecture plays a crucial role as it involves the existence of these invariants. A purpose of this note is to show how the conjecture arises. Because of the integral identity's nature we shall give a quick tour on theories of motivic integration, from which lead to a proof of the conjecture for ground fields algebraically closed of characteristic zero. IntroductionHistorically, Thomas in his thesis and his paper [31] introduced an invariant for a 3-dimensional Calabi-Yau manifold M as a counting invariant of coherent sheaves on M , which analogizes the Casson invariant on a real 3-dimensional manifold. This kind of invariant was then named after him and his advisor Donaldson. According to [16], the moduli space M of coherent sheaves on M can be locally presented as the critical locus of a holomorphic Chern-Simons functional f . By this, a Donaldson-Thomas invariant for M can be described as an integral of the Behrend function over the moduli space M. In view of [1], the value of the Behrend function is defined in terms of the Euler characteristic of the Milnor fiber F f of the Chern-Simons functional f , from which the Donaldson-Thomas invariant arises. In [17], replacing the Milnor fiber by a so-called motivic Milnor fiber (defined by Denef-Loeser [8] using motivic integration) Kontsevich and Soibelman study motivic Donaldson-Thomas invariants for 3-dimensional Calabi-Yau manifolds. The theory of Kontsevich and Soibelman allows, under appropriate realizations (e.g., a cohomology functor), to obtain refinements of (classical) Donaldson-Thomas invariants. The motivic Donaldson-Thomas invariants are also realized physically as "BPS invariants". Among 3-dimensional Calabi-Yau categories, the derived category of coherent sheaves on a compact (or local) 3-dimensional Calabi-Yau manifolds is a central object.Let us now give a brief review due to [17] and [18] on the direct elements in KontsevichSoibelman's theory concerning the integral identity conjecture. Let C be an ind-constructible triangulated A ∞ -category over a field k. For any strict sector V in R 2 , we consider a collection of full subcategories C V of C. Then one can construct the motivic Hall algebra H(C) as a graded associative algebra admitting for any strict sector V an element A Hall V invertible in a completion H(C V ) of H(C V ). More precisely, in terms of a countable decomposition Ob(C) = i∈I Y i into 2010 Mathematics Subject Classification. Primary 03C60, 14B20, 14E18, 14G22, 32S45, 11S80. Key words and phrases. motivic integration, formal schemes, rigid varieties, volume Poincaré series, resolution of singularity, integral identity conjecture, definable sets. where 1 S is the identity function interpreted as a counting measure (see [17, Section 6.1]). One requires that A Hall V must satisfy the factorization property that A Hall V = A Hall V 1 · A Hall V 2 , where V = V 1 V 2 and th...

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The moduli space of stable coherent sheaves via non-archimedean geometry

Jiang¹

2017

Preprint

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We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu's non-archimedean enumerative geometry in Gromov-Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson-Thomas invariants.In this paper we give the moduli construction over a non-archimedean field K. We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli of semistable coherent sheaves over the fractional non-archimedean field K. For a moduli space of stable sheaves of an algebraic variety X over an algebraically closed field κ, the analytification of such a moduli space gives an example of the non-archimedean moduli space.We generalize Joyce's d-critical scheme structure in [42] or Kiem-Li's virtual critical manifolds in [44] to the world of formal schemes, and Berkovich nonarchimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean K-analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik's motivic localization formula for the motivic Donaldson-Thomas invariants. CONTENTS YUNFENG JIANG 4. The moduli stack of non-archimedean (semi)-stable sheaves 4.1. The construction 4.2. Proof of Theorem 4.10 4.3. The compactness property Part II: 5. Introduction to motivic Donaldson-Thomas theory 5.1. Donaldson-Thomas invariants 5.2. Categorification of Donaldson-Thomas invariants 5.3. Motivic Donaldson-Thomas invariants and derived schemes 5.4. Contribution of this work 5.5. Outline 6. d-critical formal schemes 6.1. Definitions and Results 6.2. Proof of main theorem 6.3. d-critical non-archimedean K-analytic spaces 6.4. G m -equivariant d-critical formal schemes 7. Motivic localization of Donaldson-Thomas invariants 7.1. Grothendieck group of varieties. 7.2. Motivic integration on rigid varieties 7.3. Global motive of oriented formal d-critical schemes 7.4. Global motive for oriented d-critical non-archimedean analytic spaces 7.5. Maulik's motivic localization formula under the G m -action References J S,V,g,j R,U, f ,i : i ‹ (K b2U )| RXS Ñ j ‹ (K b2 V )| RXS such that if Φ : (R, U, f , i) ã Ñ (S, V, g, j) and Ψ : (S, V, g, j) ã Ñ (T, W, h, k) are embeddings of formal critical charts, then J

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A proof of the integral identity conjecture, II

Cited by 3 publications

References 11 publications

Euler reflexion formulas for motivic multiple zeta functions

Euler reflexion formulas for motivic multiple zeta functions

A Short Survey on the Integral Identity Conjecture and Theories of Motivic Integration

The moduli space of stable coherent sheaves via non-archimedean geometry

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