Geometry on Arc Spaces of Algebraic Varieties

By Jan Denef, François Loeser. 2000

304 citations recorded by the source corpus.

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1d46fa05ee913e9bed2dd7fe759517e0221abc3c

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In Algebra and Number Theory

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Citations in the corpus

Motivic integration on smooth rigid varieties and invariants of degenerations (2001, methodology): “In the last years, motivic integration has been shown to be a quite powerful tool in producing new invariants in birational geometry of algebraic varieties over a fieldk, say, of characteristic zero (see [18], [1], [3], [10], [11], [12]).”
Elliptic genera, real algebraic varieties and quasi-Jacobi forms (2009, background): “…characteristic e ( ] X/G ) of a crepant resolution of the quotient X/G of a complex manifold X by an action of a ﬁnite group G and the data of the action on X : (9) e ( ] X/G ) = X g,h,gh = hg e ( X g,h ) A reﬁnement of the relation (9) for the Hodge numbers and motives is given in [2], [18] [39].”
Motivic integration and the Grothendieck group of pseudo-finite fields (2002, background): “For more applications and references we refer to the survey papers [11] and [21].”
Motivic Serre invariants, ramification, and the analytic Milnor fiber (2007, background): “J (defined in Section 4) is Galois with group � mJ(k). By [ 23 ], Theorem 3.3.1, we have”
Twisted jets, motivic measures and orbifold cohomology (2001, background): “ion, we would like to review the theory of motivic measures, developed byKontsevich [Kon95], Batyrev [Bat99b], Denef andLoeser [DL99a], [DL99b]. I mention [Cra99] for a nice introduction and [Loo00], [DL00] for surveys. 1.1. Completing Grothendieck rings. Let us ﬁrst construct the ring in which moti”
A survey of characteristic classes of singular spaces (2005, background): “Once more this is an invariant of the pair (X, D), not depending on the resolution π by the transformation rule! In the language of [DL1, DL2 , DL3] it is for D = 0 just the “motivic Gorenstein volume of the arc space L(X)” of the singular space X, i.e.”
A trace formula for rigid varieties, and motivic Weil generating series for formal schemes (2007, background): “We show that, modulo normalization, its Weil generating series coincides with Denef and Loeser’s motivic zeta function associated to f , and its motivic volume coincides with the motivic nearby cycles [21].”
Arc spaces, motivic integration and stringy invariants (2004, methodology): “Here we gather a collection of various results, which were obtained after the redaction of the survey paper [ DL8 ].”
The universal Euler characteristic for varieties of characteristic zero (2001, background): “This group and its relative and equivariant analogues (to be more precise, certain localizations and completions) are the natural value groups for motivic integrals, as explained, e.g., by Looijenga in [Loo02] and by Denef and Loeser in [DL01].”
Motivic zeta functions of abelian varieties, and the monodromy conjecture (2009, background): “To be precise, Denef and Loeser’s conjecture is a bit stronger than Conjecture 2.5, because it is stated for the “monodromic” motivic zeta function which carries an additional action of the pro-k -group (cid:11) μ of roots of unity [12, 3.2.1].”
Motivic degree zero Donaldson–Thomas invariants (2009, background): “If f : M → C is a regular function on a smooth variety and Z = {df = 0} is its scheme theoretic degeneracy locus, then there is a natural virtual motive (Definition 1.13) given by [Z]vir = −L − dimM 2 [ϕf ], where [ϕf ] is the motivic vanishing cycle defined by Denef-Loeser [10, 24] and L is the Lef”
Monodromy conjecture for some surface singularities (2002, background): “In [8], J. Denef and F. Loeser gave an intrinsic definition of Ztop,0(f,s ) using arc spaces and the motivic zeta function – see also [ 10 ] and the Seminaire Bourbaki talk of E. Looijenga [24].”