## Geometry on arc spaces of algebraic varieties.(English)Zbl 1079.14003

Casacuberta, Carles (ed.) et al., 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10–14, 2000. Volume I. Basel: Birkhäuser (ISBN 3-7643-6417-3/hbk; 3-7643-6419-X/set). Prog. Math. 201, 327-348 (2001).
Let $$k$$ be an algebraically closed field of characteristic zero, and let $$X$$ be an algebraic variety defined over $$k$$. The intuitive notion of an $$n$$-order jet of an arc on $$X$$ can be expressed as a morphism from the scheme $${\text{Spec}}\, k[[t]]/(t^{n+1})$$ to $$X$$ over $$k$$. A $$0$$-jet is just a $$k$$-point of $$X$$, a $$1$$-jet is a tangent vector to $$X$$ in some $$k$$-point, and, as soon as $$n\to \infty$$, jets are getting closer to arcs on $$X$$. In a more functorial way, the scheme $${\mathcal L}_n(X)$$ of $$n$$-order jets represents the functor sending any $$k$$-algebra $$R$$ into the set $$\operatorname{Hom}_k({\text{Spec}}\, R[t]/(t^{n+1}),X)$$. The projective limit $${\mathcal L}(X)$$ of the schemes $${\mathcal L}_n(X)$$ can be viewed then as the space of (germs of) arcs on $$X$$. The morphism $${\mathcal L}_1(X)\to {\mathcal L}_0(X)=X$$ is the tangent bundle on $$X$$.
Let $${\text{Var}}$$ be the category of algebraic varieties over $$k$$. A generalized Euler characteristic is a map $$\chi$$ from $${\text{Var}}$$ to a commutative ring $$R$$ satisfying the following properties: $$\chi (X)=\chi (Y)$$ if $$X\cong Y$$, $$\chi (X)=\chi (Y)+\chi (X-Y)$$ for a Zariski closed subvariety $$Y$$ in $$X$$, and $$\chi (X\times Y)=\chi (X)\cdot \chi (Y)$$ for any two varieties $$X$$ and $$Y$$ over $$k$$. If $$k=\mathbb C$$ and $$R=\mathbb Z$$, the usual topological Euler characteristic $$\chi _{\text{top}}$$ is a particular case of the above general notion. Another example of a generalized Euler characteristic, called Hodge polynomial $$\chi _{\text{hp}}$$, can be defined by dimensions $$h_{p,q}^i$$ of $$(p,q)$$-components of the mixed Hodge structure on singular cohomology $$H^i_c(X,\mathbb C)$$ with compact support of $$X$$: $X\mapsto \chi _{\text{hp}}(X)= \sum _{i,p,q}(-1)^ih^i_{p,q}u^pv^q.$
The last characteristic can be refined using the Grothendieck group $$K_0({\text{HS}})$$ of the abelian category of Hodge structures HS. Given a mixed Hodge structure $$H$$ with weight filtration $$W_{\bullet }H$$ one has an element $$[H]=\sum _m[{\text{Gr}}_m^WH]$$ in $$K_0({\text{HS}})$$. Then the Hodge characteristic of a variety $$X$$ is given by the formula: $X\mapsto \chi _{h}(X)= \sum _{i\in \mathbb Z}(-1)^i[H^i_c(X,\mathbb C)] \in K_0({\text{HS}}) .$
The connection with arc spaces is now as follows: if $$\chi :{\roman {Var}}\to R$$ is any generalized Euler characteristic, then the series $\sum _{n\geq 0}\chi (\pi _n({\mathcal L}(X)))T^n$ is rational in the ring $$R[[T]]$$, where $$\pi _n:{\mathcal L}(X)\to {\mathcal L}_n(X)$$ is the $$n$$th canonical projection.
Moreover, one can construct a universal Euler characteristic and to prove the analogous result in the universal situation. Let $$K_0({\text{Var}})$$ be the Grothendieck group of algebraic varieties over $$k$$ with relation $$[X]=[Z]+[X-Z]$$ for Zariski closed $$Z$$ in $$X$$ and the product induced by products of varieties. Then any generalized Euler characteristic factors through the map $${\text{Var}}\to K_0({\text{Var}})$$ sending any variety $$X$$ into the corresponding generator $$[X]$$ in $$K_0({\roman {Var}})$$. Let $${\mathcal M}$$ be the ring obtained from $$K_0({\text{Var}})$$ by inverting the object $$\mathbb L=[\mathbb A^1]$$. One can show that the series $\sum _{n\geq 0}[\pi _n({\mathcal L}(X))]T^n$ is rational in the ring of formal power series $${\mathcal M}[[T]]$$ [see J. Denef and F. Loeser, Invent. Math. 135, No. 1, 201–232 (1999; Zbl 0928.14004)].
One of the ingredients in the proof of the above rationality is the notion of the motivic integration due to Kontsevich. Let $$X$$ be a variety of pure dimension $$d$$ and let $$A$$ be a constructible subset in the space $${\mathcal L}(X)$$, i.e. $$A=\pi _n^{-1}(B)$$ for some $$n$$ and constructible set $$B$$ in $${\mathcal L}_n(X)$$. Let also $$F^m{\mathcal M}$$ be a subgroup generated by $$[X]{\mathbb L}^{-i}$$ with $$\dim (X)\leq i-m$$. The filtration $$F^m$$ gives the completion $$\hat {\mathcal M}$$ of $$\mathcal M$$. Most of important Euler characteristics, including the topological Euler characteristic and Hodge characteristic, factor through the image $$\overline {{\mathcal M}}$$ of $$\mathcal M$$ in the completion $$\widehat {{\mathcal M}}$$. One can show that the limit $\mu (A):=\lim _{n\to \infty }[\pi _n(A)] {\mathbb L}^{-(n+1)d}$ exists in the completed ring $$\widehat {{\mathcal M}}$$. This gives a $$\sigma$$-additive (motivic) measure on the Boolean algebra of constructible subsets of $${\mathcal L}(X)$$. If $$X$$ is a nonsingular variety, then the motivic volume $$\mu ({\mathcal L}(X))$$ is equal to the element $$[X]{\mathbb L}^{-d}$$ in $$\widehat {{\mathcal M}}$$.
The following result plays an important role in applications. If $$f:Y\to X$$ is a proper birational morphism of varieties, then the motivic measure $$\mu (A)$$ of any constructible subset $$A$$ of $${\mathcal L}(X)$$ can be expressed as an integral over $$f^{-1}(A)$$ with respect to the measure $$\mu$$ of the power of the object $$\mathbb L$$ depending on the pull-back of the sheaf $$\Omega _X^d$$, where $$d=\dim (X)$$. In more precise terms this can be written as the generalized Kontsevich’s change of variables formula for motivic integration, see Lemma 3.3 in the above quoted paper.
For example, let $$X$$ and $$Y$$ be two Calabi-Yau complex manifolds with canonical sheaves $$\omega _X$$ and $$\omega _Y$$ respectively. Using motivic integration Kontsevich proved that the cohomology of $$X$$ and $$Y$$ have the same Hodge structures, if $$X$$ is birationally equivalent to $$Y$$. The rough scheme of the proof is as follows. There exists a complex manifold $$Z$$ with two birational morphisms $$f_X:Z\to X$$ and $$f_Y:Z\to Y$$, and that the pull-back $$f_X^*\omega _X$$ is proportional to $$f_Y^*\omega _Y$$. Using the change of variables formula one can show that the motivic volumes $$\mu ({\mathcal L}(X))$$ and $$\mu ({\mathcal L}(Y))$$ are equal to the same integral on the arc space $${\mathcal L}(Z)$$. Thus, $$[X]{\mathbb L}^{-d}=[X]{\mathbb L}^{-d}$$, where $$d$$ is the dimension of $$X$$ and $$Y$$, whence $$[X]=[Y]$$ in $$\overline {{\mathcal M}}$$.
The authors also describe another applications of the motivic integration on arc spaces to Euler characteristics, Thom-Sebastiani Theorem and motivic Poincaré series. The application to the McKay correspondence is represented in [J. Denef and F. Loeser, Compos. Math. 131, No. 3, 267–290 (2002; Zbl 1080.14001)].
For the entire collection see [Zbl 0972.00031].

### MSC:

 14B05 Singularities in algebraic geometry 14B10 Infinitesimal methods in algebraic geometry 14A15 Schemes and morphisms 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)

### Citations:

Zbl 0928.14004; Zbl 1080.14001
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