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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].

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)