ARC SPACES AND EQUIVARIANT COHOMOLOGY

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1 ARC SPACES AND EQUIVARIANT COHOMOLOGY DAVE ANDERSON AND ALAN STAPLEDON Abstract. We present a new geometrc nterpretaton of equvarant cohomology n whch one replaces a smooth, complex G-varety X by ts assocated arc space J X, wth ts nduced G-acton. Ths not only allows us to obtan geometrc classes n equvarant cohomology of arbtrarly hgh degree, but also provdes more flexblty for equvarantly deformng classes and geometrcally nterpretng multplcaton n the equvarant cohomology rng. Under approprate hypotheses, we obtan explct bjectons between Z-bases for the equvarant cohomology rngs of smooth varetes related by an equvarant, proper bratonal map. We also show that self-ntersecton classes can be represented as classes of contact loc, under certan restrctons on sngulartes of subvaretes. We gve several applcatons. Motvated by the relaton between self-ntersecton and contact loc, we defne hgher-order equvarant multplctes, generalzng the equvarant multplctes of Bron and Rossmann; these are shown to be local sngularty nvarants, and computed n some cases. We also present geometrc Z-bases for the equvarant cohomology rngs of a smooth torc varety (wth respect to the dense torus) and a partal flag varety (wth respect to the general lnear group). Contents 1. Introducton 2 2. Equvarant cohomology 5 3. Arc spaces and jet schemes 7 4. Equvarant geometry of jet schemes Jet schemes and equvarant cohomology Multplcaton of classes I Multplcaton of classes II Hgher-order multplctes Example: smooth torc varetes Example: determnantal varetes and GL n Fnal remarks 33 References 33 Date: August 3, Mathematcs Subject Classfcaton. 55N91, 14E99. Ths work was ntated and partally completed whle the authors were graduate students at the Unversty of Mchgan. A.S. was funded n part wth assstance from the Australan Research Councl project DP at Sydney Unversty, Chef Investgator Professor G I Lehrer. Part of ths work was completed durng the perod D.A. was employed by the Clay Mathematcs Insttute as a Lftoff Fellow. D.A. was also partally supported by NSF Grants DMS and DMS

2 2 DAVE ANDERSON AND ALAN STAPLEDON 1. Introducton Let X be a smooth complex algebrac varety equpped wth an acton of a lnear algebrac group G. In ths artcle, we consder two constructons assocated to ths stuaton. The equvarant cohomology rng HG X s an nterestng and useful object encodng nformaton about the topology of X as t nteracts wth the group acton; for example, fxed ponts and orbts are relevant, as are representatons of G on tangent spaces. The arc space of X s the scheme J X parametrzng morphsms SpecC[[t]] X; ths constructon s functoral, so J G s a group actng on J X. Except when X s zero-dmensonal, J X s not of fnte type over C, but t s a pro-varety, topologzed as a certan nverse lmt. Due to ther connectons wth sngularty theory [16, 18, 37] and ther central role n motvc ntegraton [13, 33], arc spaces have recently proved ncreasngly useful n bratonal geometry. The present work stems from a smple observaton: The projecton J X X s a homotopy equvalence, and s equvarant wth respect to J G G, so there s a canoncal somorphsm HG X = H J J G X (Lemma 2.1). Very broadly, our vew s that nterestng classes n HG X arse from the J G-equvarant geometry of J X. The purpose of ths artcle s to ntate an nvestgaton of the nterplay between nformaton encoded n HG X and n J X. The phlosophy we wsh to emphasze s motvated by analogy wth two notons from ordnary cohomology of (smooth) algebrac varetes. Frst, nterestng classes n H 2k X come from subvaretes of codmenson k. We seek nvarant subvaretes of codmenson k to correspond to classes n HG 2kX. Snce H GX typcally has nonzero classes n arbtrarly large degrees, however, X must be replaced wth a larger n fact, nfnte-dmensonal space. The tradtonal approach to equvarant cohomology, gong back to Borel, replaces X wth the mxng space EG G X, whch does not have a G-acton; we wll nstead study J X, whch s ntrnsc to X and on whch G acts naturally. The second general noton s that cup product n H X should correspond to transverse ntersecton of subvaretes. In the C category, of course, ths s precse: any two subvaretes can be deformed to ntersect transversely, and the cup product s represented by the ntersecton. On the other hand, X often has only fntely many G-nvarant subvaretes, so n the equvarant settng, no such movng s possble wthn X tself. Replacng X wth J X, one gans much greater flexblty to move nvarant cycles. A new and remarkable feature of ths approach s that n the case when G s the trval group, one obtans nterestng results for both the ordnary cohomology rng of X and the geometry of the arc spaces of subvaretes of X. Our frst man theorem addresses the frst noton, and says that under approprate hypotheses, J G-orbts n J X determne a bass over the ntegers: Theorem 5.7. Let G be a connected lnear algebrac group actng on a smooth complex algebrac varety X, wth D X a G-nvarant closed subset such that G acts on X D wth unpotent stablzers. Suppose J X J D = j U j s an equvarant affne pavng, n the sense of Defnton 5.6. Then H G X = j Z [U j ].

3 ARC SPACES AND EQUIVARIANT COHOMOLOGY 3 When G s a torus, the condton that G act on X D wth unpotent stablzers s a generc freeness hypothess, and s automatc n many cases of nterest. (When G s trval, ths reduces to a well-known fact about affne pavngs; see, e.g., [22, Appendx B, Lemma 6].) Snce arc spaces are well-suted to the study of the bratonal geometry of X, one should also consder proper equvarant bratonal maps f : Y X. When X satsfes the condtons of Theorem 5.7 and the pavng s compatble wth f, we establsh a geometrc bjecton between Z-bases of HG X and H GY (Corollary 5.10). Ths result s new even n the case when G s trval. In Sectons 6 and 7, we address the second noton, and relate the cup product n HG X to ntersectons n J X. The man results of these sectons (Theorem 6.1 and Theorem 7.3) say that under certan restrctons on the sngulartes of G-nvarant subvaretes, products of ther equvarant cohomology classes are represented by mult-contact loc n the arc space of X. (Basc facts about contact loc are revewed n 3.) Throughout ths artcle, we make frequent use of the jet schemes J m V = Hom(SpecC[t]/(t m+1 ),V ), whch may be consdered fnte-dmensonal approxmatons to the arc space J V. A key specal case of our theorems about multplcaton says that gven a G-nvarant subvarety V X, provded ts sngulartes are suffcently mld, we have an equalty (1) [V ] m+1 = [J m V ] as classes n H G X = H J mg (J mx). Ths specal case may be summarzed a lttle more precsely as follows: Corollary (of Theorems 6.1 and 7.3). Let G be a connected reductve group, and fx an nteger m > 0. (a) Assume X G s fnte, the natural map ι : H G X H G XG s njectve, and V X s an equvarant local complete ntersecton (see 6), wth codm(j m V,J m X) = (m + 1)c. Then [V ] m+1 = [J m V ]. (b) Assume V X s a connected G-nvarant subvarety of codmenson c, wth codm(sng(v ),X) > (m + 1)c. Then [V ] m+1 = [J m Sm(V )]. When G s a torus, the assumptons on the fxed locus n Part (a) are part of a standard package of hypotheses for localzaton theorems n equvarant cohomology; for more general groups, see Remark 6.8. Note that the statement n Part (b) apples n partcular to any smooth subvarety V X. Informaton about the sngulartes of V s encoded n the geometry of ts jet schemes, but these spaces are notorously dffcult to compute. In fact, almost nothng s known about them, except when V s a local complete ntersecton [18, 37, 38] n whch case the sequence of dmensons {dm J m V } m 0 determnes the log canoncal threshold of V [38, Corollary 0.2] or when V s a determnantal varety [34, 48]. In partcular, the correspondng class [J m V ] HG X s an mportant nvarant. When X = A d and G = (C ) r, ths class s the mult-degree of J m V (see Remark 6.4, [36, Chapter 8]). Self-ntersecton s perhaps the most dffcult part of ntersecton theory to nterpret geometrcally, because t requres some verson of a movng lemma. From an

4 4 DAVE ANDERSON AND ALAN STAPLEDON ntersecton-theoretc pont of vew, Equaton (1) gves a new geometrc nterpretaton of the self-ntersecton [V ] m+1, even n the case when G s trval and V s smooth. From the perspectve of sngularty theory, ths gves a hghly non-trval calculaton of the class [J m V ] under sutable condtons. The above Corollary mples a relatonshp between the falure of Equaton (1) and the sngulartes of V. To measure ths dscrepancy, n 8 we ntroduce hgher-order equvarant multplctes. These generalze the equvarant multplctes consdered by Rossmann [43] and Bron [9], among others; the latter have been used to study sngulartes of Schubert varetes [9, 6.5], and are related to Mnkowsk weghts on fans [31]. We prove the hgher-order multplctes are ntrnsc to V (Theorem 8.4), and apply our man results to relate them wth the (0 th -order) multplctes of Bron. Our ntal motvaton for ths work came from the theory of torc varetes. By a theorem of Ish, orbts of generc arcs n a torc varety X are parametrzed by the same set whch naturally ndexes a Z-bass for HT X, namely, ponts n the lattce N of one-parameter subgroups of T. In 9, we gve a geometrc nterpretaton of ths bjecton by applyng our results to extend t to an somorphsm of rngs (Corollary 9.3), reprovng the well-known fact that the equvarant cohomology of a smooth torc varety s somorphc to the Stanley-Resner rng of the correspondng fan. We expect ths ntrgung pcture to extend to a relaton between equvarant orbfold cohomology of torc stacks and spaces of twsted arcs n the sense of Yasuda [47] (see 11). As another applcaton, we consder the acton of GL n on n n matrces by left multplcaton. In 10, we show that Theorem 5.7 apples to ths stuaton, usng a pavng defned n terms of contact loc wth certan determnantal varetes (Corollary 10.5). Usng our results concernng the behavor of equvarant cohomology under bratonal maps (Corollary 5.10), we then deduce an arc-theoretc bass for the GL n -equvarant cohomology of a partal flag varety (Corollary 10.10). In the case when V s a determnantal varety cut out by maxmal mnors, Košr and Sethuraman proved that the jet schemes J m V are rreducble [34, Theorem 3.1]. The hypotheses for Theorems 6.1 and 7.3 fal for these subvaretes determnantal varetes are generally not l.c.., and they have large sngular sets but the conclusons appear to hold (Conjecture 10.6); t would be nterestng to have a more general framework whch explans ths. An ntrgung consequence of Equaton (1) and the more general Theorems 6.1 and 7.3 s that they allow us to conjecture, and prove n some cases, formulas for the mult-degrees of J m V (Conjecture 10.6, Remark 10.9) a calculaton whch Macaulay 2 can perform n very few examples. In the theory of equvarant cohomology, one often chooses fnte-dmensonal algebrac approxmatons to the mxng space; see, e.g., [24, 2]. (Ths approach was used by Totaro, and further developed by Eddn and Graham, to defne an algebrac theory of equvarant Chow groups.) In ths context, one may attempt to fnd representatves for classes n HG X va subvaretes of the approxmaton space (cf. [9, 2.2]) or deform to transverse poston to compute products (cf. [3]). As mentoned above, our approach uses the jet schemes J m X as fnte-dmensonal approxmatons to J X. These seem to be unrelated to the mxng space approxmatons; as wth the arc space, they have the advantages of beng ntrnsc to X and carryng large group actons.

5 ARC SPACES AND EQUIVARIANT COHOMOLOGY 5 Equvarant classes n jet schemes have also been studed by Bércz and Szenes [5], from a somewhat dfferent pont of vew. Our results overlap n a smple specal case. They consder the space J d (n,k) = Hom(Spec C[t 1,...,t n ]/(t 1,...,t n ) d+1, A k ), and compute the classes of contact loc Cont d ({0}). In general, ths s qute complcated, but n our case, when n = 1, the class n queston s c d k H GL k J d (1,k) = Z[c 1,...,c k ]. Ths s also an easy case of Conjecture 10.6 (see Remark 10.9(3)). Arc spaces have also been used by Arkhpov and Kapranov to study the quantum cohomology of torc varetes [2]. There may be an nterestng relaton between ther pont of vew and ours, but we do not know a drect connecton. For the convenence of the reader, we nclude bref summares of basc facts about equvarant cohomology ( 2) and jet schemes ( 3), together wth references. In 4, we prove a techncal fact about stablzers (Proposton 4.5) whch s used n the proof of Theorem 5.7. The man results and applcatons descrbed above are contaned n We conclude the paper wth a short dscusson of questons and projects suggested by the deas presented here. Notaton and conventons. All schemes are over the complex numbers. For us, a varety s a separated reduced scheme of fnte type over C, assumed to be puredmensonal but not necessarly rreducble. Throughout, G wll be a connected lnear algebrac group over C, and X wll be a G-varety. Unless otherwse ndcated, cohomology wll be taken wth Z coeffcents, wth respect to the usual (complex) topology. Acknowledgements. The authors would lke to thank Mrcea Mustaţǎ for several enlghtenng dscussons, and Mark Haman for suggestng that the calculaton at the end of 10 should generalze from projectve space to all partal flag varetes. We also thank Sara Blley, Bll Fulton, and Rchárd Rmány for helpful comments. 2. Equvarant cohomology We refer the reader to [24] or [10] for an ntroducton to equvarant cohomology, as well as proofs and detals. Here we collect the basc propertes we wll need, and gve a few llustratve examples. As always, G s a connected lnear algebrac group actng on the left on X. 1 A map f : X X s equvarant wth respect to a homomorphsm ϕ: G G f f(g x) = ϕ(g) f(x) for all g G, x X. Equvarant cohomology s contravarant for equvarant maps: one has f : H G X H G X. The followng two facts play a key role n our arguments: Lemma 2.1. Suppose X X s equvarant wth respect to G G, and suppose both maps nduce (weak) homotopy equvalences. Then the nduced map HG X HG X s an somorphsm. 1 By defnton, H G X s the sngular cohomology of the Borel mxng space EG G X; equvalently, t s the cohomology of the quotent stack [G\X]. The reader may consult one of the above references for a dscusson of ths constructon.

6 6 DAVE ANDERSON AND ALAN STAPLEDON (In most of our applcatons of Lemma 2.1, both maps wll be locally trval fber bundles wth contractble fbers here both the hypothess and concluson are easly verfed.) Lemma 2.2. The equvarant cohomology of an orbt s descrbed as follows: for a closed subgroup G G, one has H G (G/G ) = H G (pt). Example 2.3. For a representaton V of G, one has H G V = H G (pt). Example 2.4. If G s contractble, then HG (pt) = Z. When X s smooth, a closed G-nvarant subvarety Z X of codmenson c defnes a class [Z] n HG 2cX. If Z 1,...,Z k denote the rreducble components of Z, then [Z] = [Z 1 ] + + [Z k ]. H 2 G An equvarant vector bundle V X has equvarant Chern classes c G (V ) n X, wth the usual functoral propertes of Chern classes. Example 2.5. An equvarant vector bundle on a pont s smply a representaton of G, so one has correspondng Chern classes c G (V ) H G (pt). For V = Cn, wth GL n actng by the standard representaton, the Chern classes c = c G (V ) freely generate HGL n (pt). A key feature of equvarant cohomology s that HG X s canoncally an algebra over HG (pt), va the constant map X pt. In contrast to the non-equvarant stuaton, HG (pt) s typcally not trval. Example 2.6. If T = (C ) n s a torus wth character group M = Z n, then HT (pt) = Sym M = Z[t 1,...,t n ]. The ncluson (C ) n GL n nduces an ncluson H GL n (pt) = Z[c 1,...,c n ] Z[t 1,...,t n ], sendng c to the th elementary symmetrc functon n t. We wll use equvarant Borel-Moore homology H G X as a techncal tool; see [15, p.605] or [11, Secton 1] for some detals. The man facts are analogous to the non-equvarant case, for whch a good reference s [22, Appendx B]; we summarze them here. If X has (pure) dmenson d, then H G X = 0 for > 2d and HG 2d X = Z, wth one summand for each rreducble component of X. In contrast to the nonequvarant case, H G X may be nonzero for arbtrarly negatve. If X s smooth of dmenson d, then H G X = H 2d G X. Borel-Moore homology s covarant for equvarant proper maps and contravarant for equvarant open nclusons. For Z X a G-nvarant closed subvarety of codmenson c, there s a fundamental class [Z] n H G 2d 2cX. More generally, f Z X s any G-nvarant closed subset, wth U = X Z the open complement, there s a long exact sequence H G Z H G X H G U H G 1Z. Defnton 2.7. A d-dmensonal varety X has trval equvarant Borel-Moore homology f H G X = { Z f = 2d; 0 otherwse.

7 ARC SPACES AND EQUIVARIANT COHOMOLOGY 7 Example 2.8. For us, the man examples of such varetes arse as follows. An affne famly of G-orbts s a smooth map S A n of G-varetes, wth G actng trvally on A n, such that there s a secton s: A n S, and the map G A n S, (g,x) g s(x) s smooth and surjectve. In other words, as a smooth scheme over A n, S s the geometrc quotent of the group scheme G = G A n by a closed subgroup scheme H over A n, so we may wrte S = G/H. When H A n has contractble fbers.e., the stablzers (n G) of ponts n S are contractble subgroups the projecton S A n s a (Serre) fbraton, by [35, Corollary 15()]. It follows that H G (S) = H G (G/H 0) = Z, where G/H 0 S s the fber over 0 A n. Snce S s smooth, we conclude that S has trval Borel-Moore homology. The followng s an equvarant analogue of [22, Appendx B, Lemma 6]: Lemma 2.9. Suppose X has a fltraton by G-nvarant closed subvaretes X s X s 1 X 0 = X such that each complement U = X X +1 has trval equvarant Borel-Moore homology. Then, for 0 k < codm(x s,x), we have H G 2d 2k X = Z [U ] codm U =k and H G 2d 2k+1X = 0. Consequently, f X s smooth we have HG 2k X = Z [U ] codm U =k and H 2k 1 G X = 0, for 0 k < codm(x s,x). We omt the proof, whch proceeds exactly as n the non-equvarant case (usng nducton and the long exact sequence). We wll also need a slght refnement, whose proof s mmedate from the long exact sequence: Lemma Let X 0 X be a G-nvarant open subset. Then the nduced map H G k X H G k X 0 s an somorphsm for 2d k > 2dm(X X 0 ) Arc spaces and jet schemes In ths secton, we revew some aspects of the theory of arc spaces and jet schemes, and set notaton for the rest of the paper. We refer the reader to [37] and [17] for more detals. Let X be a scheme over C of fnte type. The m th jet scheme of X s a scheme J m X over C whose C-valued ponts parameterze all morphsms SpecC[t]/(t m+1 ) X. For example, J 0 X = X and J 1 X = TX s the total tangent space of X. In what follows, we wll often dentfy schemes wth ther C-valued ponts. For m n, the natural rng homomorphsm C[t]/(t m+1 ) C[t]/(t n+1 ) nduces truncaton morphsms π m,n : J m X J n X, and we wrte π m = π m,0 : J m X X.

8 8 DAVE ANDERSON AND ALAN STAPLEDON The ncluson C C[t]/(t m+1 ) nduces a morphsm Spec C[t]/(t m+1 ) SpecC, and hence a morphsm s m : X J m X, called the zero secton, wth the property that π m s m = d. The truncaton morphsms π m,m 1 : J m X J m 1 X form a projectve system whose projectve lmt s a scheme J X over C, whch s typcally not of fnte type. The scheme J X s called the arc space of X, and the C-valued ponts of J X parameterze all morphsms Spec C[[t]] X. For each m, there s a truncaton morphsm ψ m : J X J m X, nduced by the natural rng homomorphsm C[[t]] C[[t]]/(t m+1 ) = C[t]/(t m+1 ). Both J m and J are functors from the category of schemes of fnte type over C to the category of schemes over C, and both preserve fber squares (cf. [17, Remark 2.8]). For a morphsm f : X Y, we wrte f m : J m X J m Y for the correspondng morphsm of jet schemes. The followng lemma should be compared wth Theorem Lemma 3.1. [17, Proposton 5.12] If X s a smooth varety and V s a closed subscheme of X wth dm V < dm X, then lm codm(j mv,j m X) =. m The fundamental fact we explot n ths paper s the followng: Lemma 3.2 ([17, Corollary 2.11]). If X s a smooth varety of dmenson d, then J m X s a smooth varety of dmenson (m + 1)d, and the truncaton morphsms π m,m 1 : J m X J m 1 X are Zarsk-locally trval fbratons wth fber A d. Moreover, the projecton ψ 0 : J X X s a Zarsk-locally trval fbraton wth contractble fbers. A lttle more can be sad about the projectons, stll n the smooth case: Lemma 3.3 (see [28, Proposton 2.6]). If X s a smooth varety, the relatve tangent bundle for the truncaton map J m X J m 1 X s somorphc to π mtx. When X s sngular, J m X may not be reduced or rreducble, and may not be pure-dmensonal. However, f Sm(X) denotes the smooth locus of X, then the closure of πm 1 Sm(X) J m X s an rreducble component of dmenson (m + 1)d. Example 3.4. Let X = A n = SpecC[x 1,...,x n ]. An m-jet SpecC[t]/(t m+1 ) A n corresponds to a rng homomorphsm C[x 1,...,x n ] C[t]/(t m+1 ), and hence to an n-tuple of polynomals n t of degree at most m. We conclude that J m A n = A (m+1)n, and we wrte {x (j) 1 r,0 j m} for the correspondng coordnates. Smlarly, an arc s determned by an n-tuple of power seres over C, and J A n s an nfnte-dmensonal affne space. Example 3.5. Wth the notaton of the prevous example, f X A n s defned by equatons {f 1 (x 1,...,x n ) = = f r (x 1,...,x n ) = 0}, then an m-jet SpecC[t]/(t m+1 ) X corresponds to a rng homomorphsm C[x 1,...,x n ]/(f 1,...,f r ) C[t]/(t m+1 ).

9 ARC SPACES AND EQUIVARIANT COHOMOLOGY 9 The closed subscheme J m X J m A n = A (m+1)n s therefore defned by the equatons m m f x (j) 1 tj,..., x (j) n tj 0 mod t m+1 for 1 r. j=0 In other words, let f (k) j=0 so t s a polynomal n the varables {x (j) defned by the (m + 1)r equatons {f (k) In fact, f R = C[x (k) over C satsfyng D(x (k) be the coeffcent of t k n f ( m j=0 x(j) 1 tj,..., m j=0 x(j) n t j ), 1 r, 0 j m}. Then J m X s = 0 1 r, 0 k m}. 1 n, k 0] and D : R R s the unque dervaton ) = x (k+1), then f (k) = D k (f ) [37, p. 5]. Assume X s smooth of dmenson d. A cylnder C n J X s a subset of the arc space of X of the form C = ψm 1 (S), for some m 0 and some constructble subset S J m X. The cylnder C s called open, closed, locally closed, or rreducble f the correspondng property holds for S, and the codmenson of C s defned to be the codmenson of S n J m X. That these notons are well-defned follows from the fact that π m,m 1 s a Zarsk-locally trval fbraton wth fber A d (Lemma 3.2). A subset of J X s called thn f t s contaned n J V for some proper, closed subset V X. Lemma 3.6. [17, Proposton 5.11] Let X be a smooth varety and let C J X be a cylnder. If the complement of a dsjont unon of cylnders j C j C s thn, then lm j codm C j = and codm C = mn j codm C j. Interestng examples of cylnders arse as follows. Let V be a proper, closed subscheme of X defned by an deal sheaf I V O X, and let γ: Spec C[[t]] X be an arc. The pullback of I V va γ s ether an deal of the form (t α ), for some non-negatve nteger α, or the zero deal. In the former case, the contact order ord γ (V ) of V along γ s defned to be α; n the latter case, ord γ (V ) s nfnte by conventon, and γ les n J V J X. For each non-negatve nteger e, set Cont e (V ) = {γ J X ord γ (V ) e}, so Cont 0 (V ) = J X and Cont e (V ) = ψ 1 e 1 (J e 1V ) for e > 0. We see that Cont e (V ) s a closed cylnder and Cont e (V ) = {γ J X ord γ (V ) = e} = Cont e (V ) Cont e+1 (V ) s a locally closed cylnder. Cylnders of ths form are called contact loc. For each m e, we let Cont e (V ) m = ψ m (Cont e (V )) and Cont e (V ) m = ψ m (Cont e (V )), denote the loc of m-jets wth contact order wth V at least e and precsely e, respectvely. If subvaretes V 1,...,V s of X are specfed, along wth an s-tuple of nonnegatve ntegers e = (e 1,...,e s ), we wrte s Cont e (V ) = Cont e (V ) =1

10 10 DAVE ANDERSON AND ALAN STAPLEDON and Cont e (V ) = for the correspondng mult-contact loc. s Cont e (V ) =1 Remark 3.7. En, Lazarsfeld and Mustaţǎ [16] gave a correspondence between closed, rreducble cylnders of J X and dvsoral valuatons of the functon feld of X. We recall some results relatng arc spaces and sngulartes [37, 38, 18]. Let X be a Q-Gorensten varety, and let f : Y X be a resoluton of sngulartes such that the exceptonal locus E = E 1 E r s a smple normal crossngs dvsor. The relatve canoncal dvsor has the form K Y/X = r =1 a E, for some ntegers a, and X has termnal, canoncal, or log canoncal sngulartes f a > 0, a 0, or a 1, respectvely, for all. Theorem 3.8 ([18, Theorem 1.3]). If X s a normal, local complete ntersecton (l.c..) varety, then t has log canoncal (canoncal, termnal) sngulartes f and only f J m X s pure dmensonal (rreducble, normal) for all m 0. Remark 3.9. In general, the closure of J m Sm(X) (the jet scheme of the smooth locus) n J m X s an rreducble component of dmenson d(m+1). Thus when J m X s pure-dmensonal, ts dmenson s d(m + 1). Remark A result of Elkk [19] and Flenner [20] mples that a Gorensten varety has canoncal sngulartes f and only f t has ratonal sngulartes. Remark In fact, Mustaţǎ proves that f X s a normal, l.c.. varety wth canoncal (equvalently, ratonal) sngulartes, then J m X s l.c.., reduced, and rreducble for all m 0. Let X be a smooth varety and let V be a proper, closed subscheme. An mportant nvarant measurng the sngulartes of V s the log canoncal threshold lct(x, V ). We refer the reader to [38] for detals. Theorem 3.12 ([38, Corollary 0.2]). If X s a smooth varety and V s a proper, closed subscheme, then dm J m V lct(x,v ) = dmx max m m + 1. Moreover, the maxmum s acheved for m suffcently dvsble. The followng theorem, whch was motvated by Kontsevch s theory of motvc ntegraton, s the man ngredent n the proofs of the above results. We wll use t n Corollary Theorem 3.13 ([13]). Let f : Y X be a proper, bratonal morphsm between smooth varetes Y and X. If m 2e are non-negatve ntegers and Cont e (K Y/X ) m J m Y denotes the locus of m-jets wth contact order e wth the relatve canoncal dvsor, then the restrcton of the nduced map f m : J m Y J m X to Cont e (K Y/X ) m, f m : Cont e (K Y/X ) m f m (Cont e (K Y/X ) m ), s a Zarsk-locally trval fbraton wth fber A e.

11 ARC SPACES AND EQUIVARIANT COHOMOLOGY 11 We conclude ths secton wth a bref remark on analytfcaton. Any fnte-type C-scheme X naturally determnes a complex-analytc space X an n the sense of [25]; n partcular, one has (J m X) an. On the other hand, the analytc jet schemes of a complex-analytc space Z may be defned analogously as Jm an Z = Hom an (Spec C[t]/(t m+1 ),Z), where Spec C[t]/(t m+1 ) s consdered as an analytc space n the obvous way. Naturally, Jm an s functoral for holomorphc maps of analytc spaces. We wll use the followng lemma n the proof of Proposton 4.5. Lemma For a scheme X of fnte type over C, we have J an m (Xan ) = (J m X) an. 4. Equvarant geometry of jet schemes Let G be a lnear algebrac group actng on a smooth complex varety X. Functoralty of J m (for m n N { }) mples that J m G s an algebrac group wth an nduced acton on J m X (cf. [27, Proposton 2.6]). The man result of ths secton s Proposton 4.5, whch gves a suffcent condton for the stablzer of a pont n J m X to be contractble. Example 4.1. Let a torus T act on A n va the characters χ 1,...,χ n,.e., t (z 1,...,z n ) = (χ 1 (t)z 1,...,χ n (t)z n ). Recall that J m A n s dentfed wth n-tuples of truncated polynomals (.e., elements of C[t]/(t m+1 )). The characters also defne homomorphsms J m T J m C. Identfyng J m C wth truncated polynomals wth nonzero constant term, J m T acts on J m A n by γ (ξ 1,...,ξ n ) = (χ 1 (γ)ξ 1,...,χ n (γ)ξ n ), where the multplcaton on the RHS s multplcaton of truncated polynomals. It s also convenent to dentfy J m A n wth n (m+1) matrces, wth the entres n the kth column correspondng to the coeffcents of t k 1. Under ths dentfcaton, the zero secton T 0 J m T acts smply by scalng the th row by χ (t) C. The fxed subspace (J m A n ) T 0 s dentfed wth the rows where the correspondng character s zero; note that (J m A n ) T 0 s the m th jet scheme J m (A n ) T of the fxed locus (A n ) T. The same dscusson holds for any (possbly dsconnected) dagonalzable group H; for fnte groups, of course, there s no dfference between J m H and the zero secton. We refer the reader to [7] and [44] for basc propertes of lnear algebrac groups. In partcular, we wll need the followng fact. Lemma 4.2. Let U be a complex unpotent group, and let u denote ts Le algebra. The exponental map exp: u U s an somorphsm of complex varetes. In partcular, U s contractble. Conversely, f G s not unpotent, the quotent by ts unpotent radcal s a nontrval reductve group; such a group retracts onto a maxmal compact subgroup, so G s not contractble. In short, a lnear algebrac group s contractble f and only f t s unpotent. Let e denote the dentty element of G and, for any m 0, consder the projecton π m : J m G G and the assocated exact sequence of algebrac groups (2) 1 π 1 m (e) J m G πm G 1.

12 12 DAVE ANDERSON AND ALAN STAPLEDON The zero secton s m : G J m G (see Secton 3) dentfes J m G wth the semdrect product πm 1 (e) G. The followng lemma s stated n the Appendx n [37]. Lemma 4.3. For any m 0, the kernel π 1 m (e) of the projecton π m: J m G G s a unpotent group. The easy proof was related to us by Mustaţǎ; one uses nducton on m, the exact sequence 1 T e G πm 1 (e) π 1 m 1 (e) 1, and the fact that an extenson of a unpotent group by another unpotent group s unpotent. Lemma 4.4. Let G be a lnear algebrac group, wth maxmal torus T. Then the zero secton T 0 J m T J m G s a maxmal torus of J m G. Proof. Ths s a general fact about unpotent extensons: Suppose G = G /U, wth U G unpotent; then a torus n G s maxmal f and only f ts mage n G s maxmal. Snce every torus n G ntersects U trvally, and hence maps somorphcally to G, one mplcaton s obvous. For the other, let T G be a maxmal torus, let T G be a maxmal torus contanng the mage of T, and let H G be the premage of T, so H s solvable. Then H /U = T, so a maxmal torus of H has the same dmenson as T. It follows that T s the mage of T. (To obtan the statement of the lemma, put G = J m G and T = T 0.) Proposton 4.5. Let G be a connected lnear algebrac group actng on a smooth varety X, and let D X be a G-nvarant closed subset, wth rreducble components {D }. Assume that the acton of G on X D has unpotent stablzers. Then J m G acts on J m X J m D wth unpotent stablzers. Proof. We proceed by frst reducng to the case where G s a torus, and then to the case where X s affne space. Suppose the stablzer Γ J m G of x m J m X s not unpotent, and let γ m Γ be a nontrval semsmple element fxng x m. We wsh to show that x m les n J m D, for some rreducble component D D. Choose a maxmal torus T G, so the zero secton T 0 J m T s a maxmal torus n J m G. Snce γ m s semsmple, t les n a maxmal torus of J m G ([44, Theorem 6.4.5]). Snce all maxmal tor are conjugate ([44, Theorem 6.4.1]), there s an element c J m G such that cγ m c 1 T 0. Ths fxes c x m, and snce each rreducble component D s G-nvarant, x m les n J m D f and only f c x m does. Therefore we may assume γ m les n the torus T 0. Let H 0 = Γ T 0 be the subgroup of T 0 fxng x m ; ths s a dagonalzable group contanng γ m. Wrte H T for ts somorphc mage n G. Let x = π m (x m ) X. By assumpton, π m (γ m ) fxes x, so x les n D. Let K H be the maxmal compact subgroup. Snce H s reductve, we have an equalty of fxed pont sets X H = X K. Usng the slce theorem (see [4, I.2.1] or [32, Corollary 1.5]) together wth Lemma 3.14, we may replace X wth a K- nvarant analytc neghborhood of x, and assume X = A n wth H actng lnearly by characters χ 1,...,χ n. Snce the fxed locus (A n ) H s rreducble and contaned n D, we have (A n ) H D, for some. Usng Example 4.1, we conclude that x m (J m A n ) H 0 = J m (A n ) H J m D.

13 ARC SPACES AND EQUIVARIANT COHOMOLOGY 13 Remark 4.6. The use of the (non-algebrac) compact subgroup n the last paragraph of the proof may be slghtly unsatsfyng to some tastes. However, a nave applcaton of the natural algebrac replacement the étale slce theorem does not work, snce étale maps do not preserve rreducblty. 5. Jet schemes and equvarant cohomology In ths secton, we relate the equvarant cohomology rng HG X of a connected lnear algebrac group G actng on a smooth complex varety X of dmenson d, wth the geometry of the jet schemes J m X of X, and prove a crteron for producng a geometrc Z-bass for HG X. We wll use the followng lemma freely throughout the rest of the paper; ts proof s mmedate from Lemma 2.1 and the fact that when X s smooth, the morphsms π m : J m X X and π m : J m G G are fber bundles wth contractble fbers (Lemma 3.2). When m =, we may and wll defne HJ J G X to be HG J X. Lemma 5.1. Let X be a smooth G-varety. For any m N { }, we have somorphsms H G X H G J mx H J mg J mx. For a G-nvarant (or J m G-nvarant) closed subvarety Z J m X, we let [Z] denote the correspondng class n HG X under the somorphsm of Lemma 5.1. Observe that a closed cylnder C = ψm 1(S), for some S J mx, s G-nvarant (or J G-nvarant) f and only f S s G-nvarant (respectvely, J m G-nvarant). In ths case, t follows from Lemma 5.1 that there s a well-defned class [C] = [S] HG X. The followng lemma s a drect applcaton of Lemma 2.9 and Lemma Lemma 5.2. Let G be a connected lnear algebrac group actng on a smooth complex varety X, wth D X a G-nvarant closed subset wth rreducble components D 1,...,D t. Suppose there exsts a fltraton by J m G-nvarant closed subvaretes Z s Z 0 = J m X J m D, such that each U j = Z j Z j+1 has trval equvarant Borel-Moore homology (see Defnton 2.7). Settng k = mn{codm(z s,j m X), mn{codm(j m D,J m X)}} 1, we have H 2k G X = Z [U j ]. codm U j k Remark 5.3. Lemma 3.1 mples that lm m codm(j m D,J m X) =. In fact, Theorem 3.12 mples that codm(j m D,J m X) (m + 1)lct(X,D ), and equalty s acheved for m suffcently dvsble. In order to state our results, we ntroduce the followng notaton. Recall from Example 2.8 that a G-varety S s an affne famly of G-orbts f there s a smooth map S A n, and S s dentfed wth a geometrc quotent of G A n by some closed subgroup scheme over A n. Defnton 5.4. Let D X be a G-nvarant closed subset wth rreducble components D 1,...,D t. A locally closed cylnder C J X s an affne famly of orbts (wth respect to D) f C = ψm 1(S) for some S J mx, such that S J m D = for all, and S s an affne famly of J m G-orbts.

14 14 DAVE ANDERSON AND ALAN STAPLEDON Remark 5.5. Wth the notaton above, suppose that G acts on X D wth unpotent stablzers. By Proposton 4.5 and Lemma 4.2, the stablzer of x S J m X J md s contractble, so Example 2.8 shows that S has trval equvarant Borel-Moore homology. Moreover, for any m m, π 1 m,m (S) J m X J m D s smooth and hence has trval equvarant Borel-Moore homology by Lemma 3.2 and Lemma 2.1. Defnton 5.6. Wth the notaton of Lemma 5.2, a decomposton J X J D = j U j nto a non-empty, dsjont unon of cylnders s an equvarant affne pavng f there exsts a fltraton J D Z j+1 Z j Z 0 = J X by J G-nvarant closed cylnders n J X contanng J D such that U j = Z j Z j+1 s an affne famly of orbts. We are now ready to present our frst man theorem. Theorem 5.7. Let G be a connected lnear algebrac group actng on a smooth complex varety X, wth D X a G-nvarant closed subset such that G acts on X D wth unpotent stablzers. If J X J D = j U j s an equvarant affne pavng, then HG X = Z [U j ]. j Proof. We assume the notaton of Defnton 5.6. Fx a degree k, and note that the fltraton s ether fnte or satsfes lm j codm U j = by Lemma 3.6; therefore the set {j codm U j k} s always fnte. Let s 1 be the largest ndex n ths fnte set (so codm Z s > k by Lemma 3.6). Now choose m large enough so that Z j = ψm 1 (ψ m (Z j )) for j s, and U j = ψm 1 (S j ) for j < s, where S j J m X Jm D s an affne famly of J m G-orbts. Also choose m large enough so that 2mn{codm(J m D,J m X)} > k (see Remark 5.3), where the D are the rreducble components of D. Settng Z j = ψ m(z j ) J md, we have a fltraton of J m G- nvarant closed subvaretes Z s Z 0 = J mx J m D, such that each ψ m (U j ) = Z j Z j+1 has trval equvarant Borel-Moore homology by Remark 5.5. The result now follows from Lemma 5.2. Remark 5.8. If G acts on a smooth varety X wth a free, dense open orbt U, then G acts on U wth trval, and hence unpotent, stablzers. Applcatons of Theorem 5.7 of ths type are gven n Secton 9 and Secton 10. Remark 5.9. The smplest type of cylnder whch s an affne famly of orbts conssts of a sngle J G-orbt n J X. The exstence of an equvarant affne pavng nvolvng only cells of ths type s qute restrctve, however. Indeed, suppose X s compact and G acts freely on X D. The valuatve crteron for properness [26, Theorem II.4.7] mples that there s a bjecton between J G-orbts of J X J D and elements of the affne Grassmannan G((t))/G[[t]], and the latter s uncountable unless G s dagonalzable. Snce our noton of pavng assumes countably many orbts n fact, fntely many n any gven codmenson essentally the only examples of ths type are compactfcatons of tor,.e. torc varetes (see Secton 9).

15 ARC SPACES AND EQUIVARIANT COHOMOLOGY 15 For the remander of the secton, we wll consder a proper, equvarant bratonal map f : Y X between smooth G-varetes Y and X, for some connected lnear algebrac group G. We wll apply our results above to descrbe a method for comparng the G-equvarant cohomology of X and Y. Suppose that D X s a G-nvarant closed subset such that G acts on X D wth unpotent stablzers, and, wth the notaton of Defnton 5.6, consder an equvarant affne pavng J X J D = j U j. Recall that the relatve canoncal dvsor K Y/X on Y s the dvsor defned by the vanshng of the Jacoban of f : Y X, and that f : J Y J X denotes the morphsm of arc spaces correspondng to f. We say that the pavng s compatble wth f f f 1(U j) Cont e j (K Y/X ) for some non-negatve nteger e j and for all j. In ths case, we wll wrte e j = ord f 1 (U j ) (K Y/X). Corollary Let G be a connected lnear algebrac group and let f : Y X be a proper, equvarant bratonal map between smooth G-varetes Y and X. Let D X be a G-nvarant closed subset such that G acts on X D wth unpotent stablzers, and let J X J D = j U j be an equvarant affne pavng whch s compatble wth f. These data determne a bjecton between Z-bases of HG Y and HG X, explctly gven by H GY = j Z [f 1 (U j )], H GX = j Z [U j ]. Moreover, codm U j = codm f 1 (U j) + e j, where e j = ord f 1 (U j ) (K Y/X). Proof. The pavng of Defnton 5.6, J D Z j+1 Z j Z 0 = J X, lfts to a chan of J G-nvarant closed cylnders contanng J (f 1 (D)): f 1 (J D) = J (f 1 (D)) f 1 (Z j+1 ) f 1 (Z j ) f 1 (Z 0 ) = J Y. Here f 1(Z j) f 1(Z j+1) = f 1(U j), and f 1 (D) denotes the scheme-theoretc nverse mage of D. Fx a degree k and note that the set {j codm f 1 (U j ) k} s fnte by Lemma 3.6. Let s 1 be an ndex greater than max({j codm f 1 (U j ) k}) and max({j codm U j k}). By the proof of Theorem 5.7 and Lemma 5.2, we may choose m suffcently large such that we have a fltraton of J m G-nvarant closed subvaretes Z s Z 0 = J mx J m D, such that each ψm(u X j ) = Z j Z j+1 has trval Borel-Moore homology and U j = ψm 1 (ψm(u X j )). Moreover, f D 1,...,D t denote the rreducble components of D, then we may choose m large enough so that 2mn{codm(J m f 1 (D ),J m Y )} > k (by Lemma 3.1) and m 2e j for 0 j s 1. Consder the fltraton f 1 m (Z s) f 1 m (Z 0) = J m Y J m f 1 (D ), wth fm 1 (Z j ) f 1 m (Z j+1 ) = f m 1 (ψm(u X j )) = ψm(f Y 1 (U j )). By Theorem 3.13, the restrcton f m : fm 1(U j) U j s a J m G-equvarant, Zarsk-locally trval fbraton

16 16 DAVE ANDERSON AND ALAN STAPLEDON wth fber A e j. We conclude that fm 1(U j) has trval equvarant Borel-Moore homology and codm U j = codm f 1(U j)+e j. Usng Lemma 2.9 and Lemma 2.10, we conclude that H 2k 1 G Y = 0 and HG 2k Y = Z [f 1 (U j )]. codm U j +e j =k The result now follows from Theorem 5.7. In the succeedng two sectons, we wll gve crtera to nterpret multplcaton n the equvarant cohomology rng geometrcally. An answer to the followng queston may be very useful n provng Conjecture 10.6 (cf. Example and Remark 10.7): Queston Under sutable assumptons, can one compare the multplcaton of classes n the Z-bases of HG Y and H GX determned by Corollary 5.10? Remark The relatonshp between the graded dmensons of HG (Y ; C) and HG (X; C), and the relatve canoncal dvsor K Y/X, would be predcted by an equvarant verson of motvc ntegraton. We say that two smooth G-varetes X and Y are equvarantly K-equvalent f there s a smooth G-varety Z and G-equvarant, proper bratonal maps Z X and Z Y such that K Z/X = K Z/Y. For example, one may consder equvarantly K-equvalent torc varetes wth respect to the torus acton (cf. Secton 9). As n the non-equvarant case, one expects that f X and Y are equvarantly K-equvalent, then dm C HG (X; C) = dm C HG (Y ; C) for all 0. Queston Do there exst nterestng examples of G-equvarantly K-equvalent varetes, where G s non-trval, other than K-equvalent torc varetes? 6. Multplcaton of classes I In ths secton and the next, we use jet schemes to gve a geometrc nterpretaton of multplcaton n the equvarant cohomology rng HG X of a smooth varety X acted on by a connected lnear algebrac group G. We present two sets of results, wth dfferent assumptons on the sngulartes of subvaretes: the frst concerns local complete ntersecton varetes (treated n ths secton), and the second requres the sngular locus to be suffcently small (dscussed n the followng secton). It wll be convenent to ntroduce some termnology for ths secton. A subvarety V X s an equvarant complete ntersecton f t has codmenson r and s the scheme-theoretc ntersecton of r G-nvarant hypersurfaces n X. Smlarly, V X s an equvarant local complete ntersecton (e.l.c..) f t s a local complete ntersecton varety locally cut out by G-nvarant hypersurfaces. Of course, a G- nvarant l.c.. subvarety need not be e.l.c..: for example, the orgn n C n s not cut out by GL n -nvarant hypersurfaces (snce there are no such hypersurfaces). For a tuple of non-negatve ntegers m = (m 1,...,m s ), let λ(m) = (λ 1,...,λ s ) be the partton defned by λ = m + + m s. The man theorem of ths secton s ths: Theorem 6.1. Assume the followng: ( ) G s a connected reductve group, X G s fnte, and the natural map ι : H G X H G XG s njectve.

17 ARC SPACES AND EQUIVARIANT COHOMOLOGY 17 Consder a chan of e.l.c.. subvaretes V s V s 1 V 1 X, and a tuple m = (m 1,...,m s ) of non-negatve ntegers. If codm Cont λ(m) (V ) = s =1 m codm V, then (3) [V 1 ] m1 [V s ] ms = [Cont λ(m) (V )] Remark 6.2. In the statement of the above theorem, observe that f the hypothess codm Cont λ(m) (V ) = s =1 m codm V holds for all tuples m = (m 1,...,m s ) of non-negatve ntegers, then [Cont λ(m) (V )] = [Cont λ(m) (V )]. We wll prove Theorem 6.1 by reducng to the case of A d. The assumpton ( ) s needed only for the reducton, so we do not requre t n what follows, when X = A d. Let G act on A d and let V A d be a G-nvarant hypersurface, defned by f C[x 1,...,x n ]. Recall from Example 3.5 that J m V J m A d s defned by equatons {f (k) 0 k m} n the varables {x (k) 1 n, 0 k m}. Lemma 6.3. For 0 k m, the hypersurface V (k) := {f + f (k) = 0} J m A d s G-nvarant, and under the somorphsm of Lemma 5.1, [V (k) ] = [V ] H G Ad. Proof. The lemma s trval when k = 0, so assume k 1. Snce V s nvarant, g f = λ(g)f for some character λ: G C. Wth the notaton of Example 3.5, t follows from the defnton of the acton of G on J m A d that ( m ) ( m m ) (g f) x (k) m 1 tk,..., = λ(g)f x (k) 1 tk,...,. k=0 k=0 x (k) d tk k=0 k=0 x (k) d tk In partcular, consderng coeffcents of t k on both sdes gves g f (k) = λ(g)f (k) for 0 k m, and we conclude that V (k) s G-nvarant. Let V J m A d A 1 be defned by the equaton f + ζf (k) = 0 (where ζ s the parameter on A 1 ). Thus V A 1 s an equvarant famly of hypersurfaces n J m A d, whose fbers at ζ = 0 and ζ = 1 are V and V (k), respectvely. (The polynomals f and f (k) nvolve dfferent varables, so f + ζf (k) s never dentcally zero; hence each fber has the same dmenson.) Snce V s a hypersurface n an affne space, t follows that the projecton V A 1 s flat; ndeed, one easly checks that C[V] s torson free and hence free over C[ζ]. We conclude that [V (k) ] = [V ]. Remark 6.4. If G = (C ) r s a torus, then the equvarant cohomology class of a torus-nvarant subvarety V A d s equal to ts mult-degree [36, Chapter 8]. In ths case, t follows from the descrpton of f (k) as an terated dervaton of f n Example 3.5 that V and V (k) have the same mult-degree, mplyng the above lemma. Recall that for a tuple of non-negatve ntegers m = (m 1,...,m s ), we let λ(m) = (λ 1,...,λ s ) be the partton defned by λ = m + + m s. Proposton 6.5. Consder a chan of equvarant complete ntersecton subvaretes V s V s 1 V 1 A d,

18 18 DAVE ANDERSON AND ALAN STAPLEDON and a tuple m = (m 1,...,m s ) of non-negatve ntegers. If codm Cont λ(m) (V ) = s =1 m codm V, then [V 1 ] m1 [V s ] ms = [Cont λ(m) (V )]. Proof. We wll show that Cont λ(m) (V ) s an equvarant complete ntersecton. Fx m λ 1 1, so the equatons defnng Cont λ(m) (V ) are the same as those defnng π 1 m,λ 1 (J λ 1V ) n J m A d. It wll suffce to prove the clamed equaton n H G J ma d. For each, let r = codm V j and let f,1,...,f,r be (sem-nvarant) polynomals defnng V. Thus {f (k),j 1 j r, 0 k λ 1} defnes J λ 1V n J λ 1A d, as well as π 1 m,λ 1 (J λ 1V ). Now consder V s V s 1. Snce J λs 1V s J λs 1V s 1, we have a contanment of deals (f (k) s,j 1 j r s, 0 k λ s 1) (f (k) s 1,j 1 j r s 1, 0 k λ s 1). To cut out π 1 m,λ (J s 1 λ s 1V s ) π 1 m,λ s 1 1 (J λ s 1 1V s 1 ), then, we need the m s r s equatons {f (k) s,j 1 j r s, 0 k λ s 1} together wth the m s 1 r s 1 equatons {f (k) s 1,j 1 j r s 1, λ s k λ s 1 1}. Contnung n ths way, we obtan s =1 m r equatons defnng π 1 m,λ 1 (J λ 1V ); by hypothess, ths s the codmenson of π 1 m,λ 1 (J λ 1V ), so t s a complete ntersecton. It follows that where V (k),j the class [V (k) [ π 1 m,λ 1 (J λ 1V )] = s r λ 1 [V (k),j ], =1 j=1 k=λ 1 J m A d s the G-nvarant hypersurface defned by f (k),j. By Lemma 6.3,,j ] s ndependent of k, and snce V s a complete ntersecton, we have r (0) j=1 [V,j ] = [V ]. The proposton follows. In practce, the codmenson condton n the above proposton may be dffcult to check. It would be very nterestng to have a nce answer to the followng queston. Queston 6.6. Can one gve a geometrc crteron for the codmenson condton n Proposton 6.5 to be satsfed for all tuples m = (m 1,...,m s ) of non-negatve ntegers? In the case when V s = V 1 = V A d, we have the followng answer. Corollary 6.7. Suppose V A d s an equvarant complete ntersecton. Then [J m V ] = [V ] m+1 whenever J m V s pure-dmensonal. In partcular, f V s normal and [V ] s not nlpotent n HG X, then ths equaton holds for all m 0 f and only f V has log canoncal sngulartes.

19 ARC SPACES AND EQUIVARIANT COHOMOLOGY 19 Proof. The frst statement follows from Proposton 6.5, and the second s mmedate from Theorem 3.8. Proof of Theorem 6.1. By hypothess ( ), HG X embeds n H G XG, so t suffces to establsh the formula (3) after restrcton to a fxed pont p X G. Snce G s reductve, the slce theorem gves a G-nvarant (étale or analytc) neghborhood of p equvarantly somorphc to A d. Now apply Proposton 6.5. Remark 6.8. If one uses Q coeffcents for cohomology, the hypothess ( ) n Theorem 6.1 can be replaced by the followng: ( ) G s connected, and for a maxmal torus T G, X T s fnte and the map HT X H T XT s njectve. Moreover, we may assume that our subvaretes {V } are e.l.c. wth respect to T. Indeed, ( ) apples to T, and HG X embeds n H TX as the subrng of Weyl nvarants, by a theorem of Borel. Corollary 6.7 also extends to e.l.c.. subvaretes, usng ether hypothess ( ) or ( ). The followng varant s useful; t follows mmedately from Theorem 6.1. Corollary 6.9. Assume hypothess ( ), and let Y 1,...,Y s be nvarant subvaretes of X such that each ntersecton V = Y 1 Y s proper and e.l.c.. For a tuple m = (m 1,...,m s ) of non-negatve ntegers, f codm Cont m (Y ) = s =1 m codm Y, then [Y 1 ] m1 [Y s ] ms = [Cont m (Y )]. Example Suppose G and X satsfy ( ), and let D = D D s be a G-nvarant normal crossngs dvsor n X. Corollary 6.9 and Remark 6.2 apply, so [D 1 ] m1 [D s ] ms = [Cont m (D)] = [Cont m (D)]. 7. Multplcaton of classes II Replacng the assumpton that V X be an equvarant local complete ntersecton wth a restrcton on the dmenson of the sngular locus of V, we can prove versons of the results of the prevous secton. Throughout ths secton, G s assumed to be reductve. In what follows, we wll embed X as a smooth subvarety of J m X va the zero secton, and wrte m+1 : X X X for the dagonal embeddng of X n the (m + 1)-fold product. Lemma 7.1. There are canoncal somorphsms N X/JmX = N m+1 /X X = TX m. Proof. By the functoral defnton of jet schemes, T(J m X) = Hom(Spec C[s]/(s 2 ),J m X) = Hom(Spec C[s,t]/(s 2,t m+1 ),X), so for a closed pont x n X, a vector n T x J m X corresponds to a C-algebra homomorphsm m θ: O X,x C[s,t]/(s 2,t m+1 ), θ(y) = θ 0 (y) + ϕ (y)st, =0

20 20 DAVE ANDERSON AND ALAN STAPLEDON where θ 0 : O X,x C s the C-algebra homomorphsm correspondng to x. That θ s a C-algebra homomorphsm s equvalent to requrng that θ 0 (y)+sϕ (y) s a closed pont n T x X for 0 m. We therefore have a natural somorphsm T x J m X = T x X T x X. Moreover, dentfyng X wth the zero secton, we have an embeddng of T x X n T x J m X whose mage corresponds to the subspace where ϕ 1 = = ϕ m = 0. On the other hand, T x (X X) = T x X T x X, and T x m+1 s the mage of T x X under the dagonal embeddng n T x X T x X. Hence (wth a slght abuse of notaton) N X/JmX,x = {(ϕ 0,ϕ 1,...,ϕ m ) ϕ T x X}/{(ϕ,0,...,0) ϕ T x X}, N m+1 /X X,x = {(ϕ 0,ϕ 1,...,ϕ m ) ϕ T xx}/{(ϕ,ϕ,...,ϕ ) ϕ T x X}, and there s a natural somorphsm sendng (ϕ 0,ϕ 1,...,ϕ m ) (ϕ 0,ϕ 0 ϕ 1,...,ϕ 0 ϕ m ). One easly verfes that ths extends to a canoncal global somorphsm. For the remander of the secton we consder a chan of nvarant rreducble subvaretes V s V s 1 V 1 X, and a tuple m = (m 1,...,m s ) of non-negatve ntegers wth m s > 0. Recall that λ(m) = (λ 1,...,λ s ) denotes the partton defned by λ = m + + m s, and Cont λ(m) (V ) denotes the assocated mult-contact locus. If U = X Sng(V ), then Cont λ(m) (V ) restrcts to a smooth, rreducble cylnder n J U. The closure of ths restrcted cylnder n J X s an rreducble cylnder of codmenson m codm V whch we denote by Cont λ(m) Sm(V ). Remark 7.2. Consder a chan of nvarant smooth subvaretes V s V s 1 V 1 X, and a tuple m = (m 1,...,m s ) of non-negatve ntegers wth m s > 0. Fx m λ 1 1, so that Cont λ(m) (V ) m = π 1 m,λ 1 (J λ 1V ) n J m X. The proof of Lemma 7.1 gves a canoncal somorphsm between the normal bundle of V s, embedded va the zero secton n Cont λ(m) (V ) m, and the normal bundle of V s, embedded va the dagonal embeddng n V s V }{{} s V 1 V 1 }{{}} X {{ X }. m s tmes m 1 tmes m+1 λ 1 tmes Theorem 7.3. Let X be a smooth G-varety of dmenson d, and consder a chan of nvarant subvaretes V s V s 1 V 1 X, and a tuple m = (m 1,...,m s ) of non-negatve ntegers. We have [V 1 ] m1 [V s ] ms = [Cont λ(m) (V )] whenever mn{codm(sng(v r ),X)} > m codm(v,x). (By conventon, dm =.)