A trace formula for rigid varieties, and motivic Weil generating series for formal schemes

Nicaise, Johannes

doi:10.1007/s00208-008-0273-9

Cited by 48 publications

(118 citation statements)

References 33 publications

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“…Because of working only on affine formal schemes we shall recall the explicit construction of generic fiber in this case, following [7] and [28]. Now assume that X = Spf(A) with A a special R-algebra.…”

Section: Special Formal Schemes and Associated Rigid Varietiesmentioning

confidence: 99%

“…Furthermore, the correspondance X → X η is functorial, i.e., it defines a functor from the category of special formal R-schemes to the category of separated rigid K-varieties. A special formal R-scheme X is called generically smooth if its generic fiber X η is smooth over K.Because of working only on affine formal schemes we shall recall the explicit construction of generic fiber in this case, following [7] and [28]. Now assume that X = Spf(A) with A a special R-algebra.…”

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confidence: 99%

“…, r, are the irreducible components of X. In particular, the irreducible components of an affine special formal R-scheme Spf(A) correspond to the minimal prime ideals of A (see [28, Lemma 2.29]).Let X be a regular special formal R-scheme, i.e., O X,x is regular for any x ∈ X, and E a closed formal subscheme of X. Recall from [28] that E is a strict normal crossings divisor if, for each x in X, there exists a regular system of local parameters (x 0 , .…”

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confidence: 99%

“…In particular, the irreducible components of an affine special formal R-scheme Spf(A) correspond to the minimal prime ideals of A (see [28, Lemma 2.29]).…”

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confidence: 99%

“…Let X be a generically smooth special formal R-scheme, ω a gauge form on X η . According to [28], the volume Poincaré series of (X, ω) is defined to be an element ofWe remark that volume Poincaré series in motivic integration was firstly introduced by Nicaise and Sebag in [24] for any generically smooth separated formal R-scheme topologically of finite type together with a gauge form on its generic fiber. With the previous definition we recall a generalization of the volume Poincaré series by Nicaise in the framework of motivic integration for special formal schemes (see [28]).…”

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confidence: 99%

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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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Section: Special Formal Schemes and Associated Rigid Varietiesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…In particular, the irreducible components of an affine special formal R-scheme Spf(A) correspond to the minimal prime ideals of A (see [28, Lemma 2.29]).…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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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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Chambert-Loir¹,

Nicaise²,

Sebag³

2018

Progress in Mathematics

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Motivic zeta functions of degenerating Calabi–Yau varieties

Halle

Nicaise

2017

Math. Ann.

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Abstract. We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry.

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A trace formula for rigid varieties, and motivic Weil generating series for formal schemes

Cited by 48 publications

References 33 publications

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

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

Applications

Motivic zeta functions of degenerating Calabi–Yau varieties

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