COMPACTIFICATIONS OF SUBSCHEMES OF INTEGRAL MODELS OF SHIMURA VARIETIES

Transcription

1 COMPACTIFICATIONS OF SUBSCEMES OF INTEGRAL MODELS OF SIMURA VARIETIES KAI-WEN LAN AND BENOÎT STRO Abstract. We study several knds of subschemes of mxed characterstcs models of Shmura varetes whch admt good torodal and mnmal compactfcatons, wth famlar boundary stratfcatons and formal local structures, as f they are Shmura varetes n characterstc zero. We also generalze Koecher s prncple and the relatve vanshng of subcanoncal extensons for coherent sheaves, and Pnk s and Morel s formulae for étale sheaves, to the context of such subschemes. Contents 1. Introducton 2 2. Well-postoned subsets and subschemes of good ntegral models Background settng Well-postoned subsets and subschemes Compactfcatons of well-postoned subschemes Functoral propertes and ecke actons Vanshng of hgher drect mages, and Koecher s prncple Examples of well-postoned subsets and subschemes Pullbacks and fbers p-rank strata and ther pullbacks Newton strata and ther pullbacks Oort central leaves and ther pullbacks Ekedahl Oort strata and ther pullbacks Kottwtz Rapoport strata and ther pullbacks Supports of nearby cycles Well-postoned étale sheaves Defnton General propertes and examples Pnk s formula Mantovan s formula wth boundary terms Morel s formula 58 References Mathematcs Subject Classfcaton. Prmary 11G18; Secondary 11G15, 11F75. The frst author s partally supported by the Natonal Scence Foundaton under agreements Nos. DMS and DMS , by an Alfred P. Sloan Research Fellowshp, and by the Unversté Pars 13. The second author s partally supported by the A.N.R. (Agence Natonale de la Recherche) under the program ANR-14-CE

2 2 KAI-WEN LAN AND BENOÎT STRO 1. Introducton Integral models of Shmura varetes and ther compactfcatons have played crucally mportant roles n many recent developments n algebrac number theory. In most of these developments, t s desrable to have certan decompostons of ther specal fbers nto dsjont unons of locally closed subsets, whch allowed mathematcans to reduce or relate ther questons to some smpler buldng blocks. For example, n PEL-type cases wthout factors of type D, we have the p-rank strata and Newton strata for all models, the Oort central leaves and Ekedahl Oort strata for models above hyperspecal levels at p, and the Kottwtz Rapoport strata for models above parahorc levels at p, on the reductons modulo p of all ntegral models we consder. When the level and ramfcaton at p are mld, these are known to form stratfcatons n the precse sense that closures of strata are agan unons of strata. In general, they stll form stratfcatons n a weaker sense that sutable unons form closed subsets, whch are suffcently useful for many applcatons. On the other hand, t s also desrable to have nce total or partal compactfcatons of the ntegral models. For a long tme, t was manly the good reducton ntegral models or some parahorc varants of them whch had been consdered at all (see the ntroductons of [34], [71], and [72]). Nevertheless, n recent works by Madapus Pera (see [44]) and by one of us (see [36], [40], and [38], wth more elementary arguments), a general prncple has emerged the dffcultes n the constructon of compactfcatons and n the constructon of normal ntegral models wth nce local propertes are essentally dsjont from each other. Roughly speakng, the frst goal of ths artcle s to unformly construct good partal torodal and mnmal compactfcatons for many nce locally closed subsets or subschemes, wthout any detaled knowledge of ther local propertes, but wth a long lst of nce propertes as f they are Shmura varetes n characterstc zero. We have several motvatons for ths. Frstly, whle preparng our prevous artcle [41], we observed that the supports of nearby cycles over the ntegral models of PEL-type or odge-type Shmura varetes we consder, even n the trval coeffcent case, enjoy some ntrgung nce features near the torodal and mnmal boundary, whch make t possble to talk about good torodal and mnmal compactfcatons of such supports. (We emphasze that n PEL-type cases we do allow arbtrarly hgh levels at p, for whch no theory of local models s currently avalable.) Base on earler experence of one of us (see [73] and [75]), we soon realze that the same can be sad for several other knds of subsets or subschemes of the ntegral models we consder, ncludng the abovementoned p-rank strata, Newton strata, Oort central leaves, Ekedahl Oort strata, and Kottwtz Rapoport strata (at least n PEL-type cases). Secondly, n Boxer s recent work [11] on generalzed asse nvarants on Ekedahl Oort strata, he ntroduced the noton of well-postoned subschemes near the boundares of torodal and mnmal compactfcatons of the good reducton ntegral models constructed n [34], and used t to show that the Ekedahl Oort strata extend to affne subschemes of the mnmal compactfcatons. (In the closely related work [19], n ts current verson, they had only establshed ths last statement n the Segel or compact cases. As far as we can see, what they lacked n the other noncompact cases s exactly the well-poston property observed by Boxer.) We observed that

3 COMPACTIFICATIONS OF SUBSCEMES 3 a slghtly weaker noton than Boxer s naturally generalzes to cover many other nterestng stuatons, ncludng everythng we mentoned n the prevous paragraph. We shall call them well-postoned subsets or subschemes, from now on. Thrdly, we can generalze many useful results concernng the coherent and étale cohomology of ntegral models of Shmura varetes to the context of well-postoned subsets or subschemes these can be consdered the second goal of ths artcle. They not only have many potentally applcatons, but also clarfy what were really needed n ther orgnal proofs. For example, for coherent cohomology, we can generalze Koecher s prncple for the global sectons of canoncal extensons of automorphc bundles over partal torodal compactfcatons, and the vanshng of hgher drect mages of subcanoncal extensons of automorphc bundles under the canoncal morphsms from the partal torodal compactfcatons to the partal mnmal compactfcatons, even when they are far from normal. For étale cohomology, under some techncal assumptons, we can generalze Pnk s formula (see [64]) for the pullbacks to boundary strata of the derved drect mages of automorphc étale sheaves, under the canoncal morphsm from Shmura varetes to the mnmal compactfcatons; and also Morel s formula (see [52], [53], and [55]) for the analogues for mddle perversty extensons nstead of derved drect mages. ere s an outlne of ths artcle. In Secton 2, we ntroduce and study the notons of well-postoned subsets and subschemes of the ntegral models of PEL-type or odge-type Shmura varetes we consder, and of ther partal torodal and mnmal compactfcatons. In Secton 2.1, we revew the ntegral models we consder, and gve a qualtatve descrpton of ther torodal and mnmal compactfcatons. In Secton 2.2, we ntroduce the well-postoned subsets and subschemes, and prove some basc lemmas. In Secton 2.3, we construct the partal torodal and mnmal compactfcatons for all well-postoned subsets and subschemes, and nclude some basc consequences. For example, we show that the local propertes of the partal torodal compactfcatons are as nce as the gven well-postoned subset (wth reduced subscheme structure) or subscheme. In Secton 2.4, we show that the consderaton of well-postoned subsets and subschemes, and ther partal compactfcatons, are functoral n nature and compatble wth ecke actons. In Secton 2.5, we present the generalzatons of Koecher s prncple and the vanshng of hgher drect mages alluded above. In Secton 3, we study many examples of well-postoned subsets and subschemes. We start wth the seemngly trval examples of pullbacks and fbers n Secton 3.1. Then we proceed wth the more nterestng examples of the p-rank strata n Secton 3.2, the Newton strata n Secton 3.3, the Oort central leaves n Secton 3.4, the Ekedahl Oort strata n Secton 3.5, and the Kottwtz Rapoport strata n Secton 3.6, all n PEL-type cases (and often wthout factors of type D), by frst ntroducng them as locally closed subsets, and then showng that they are well postoned as n Secton 2.2, and hence admt partal torodal and mnmal compactfcatons as n Secton 2.3. We have chosen to present these examples n PEL-type cases, often wthout factors of type D, because the theores are most complete and well understood n these cases. The defntons we have used are not necessarly the ones of the greatest elegance, novelty, or hstorcal mportance, but rather ones that are most amenable to the consderaton of sem-abelan degeneratons near the boundary. (We apologze to any experts whose works we mght have faled to hghlght or even menton.) In Secton 3.7, we study the supports of nearby

4 4 KAI-WEN LAN AND BENOÎT STRO cycles, and show that they are well postoned, under an assumpton that s satsfed at a cofnal system of levels. The upshot s that, wth lttle knowledge beyond the defntons, we can show that many subsets or subschemes are well-postoned. Then they automatcally admt partal torodal and mnmal compactfcatons wth famlar propertes, just lke ther ambent ntegral models of Shmura varetes, and they enjoy all the nce features we have abstractly establshed n Secton 2. In Secton 4, we ntroduce and study the noton of well-postoned étale sheaves and complexes over the well-postoned subsets ntroduced n Secton 2. In Sectons 4.1 and 4.2, we gve ther defntons and study ther general propertes, together wth some examples. In Sectons 4.3, 4.4, and 4.5, we present the generalzatons of Pnk s and Morel s formulae alluded above, and also a varant of Mantovan s formula wth boundary terms (dfferent from our prevous generalzaton n [41]). We emphasze that our results apply even when we have essentally no knowledge of the well-postoned subsets etc beng consdered. For example, consder any good reducton ntegral model n PEL-type cases wthout factors of type D ntroduced by Kottwtz n [30, Sec. 5], consder the ntersecton of a Newton and an Ekedahl Oort strata, and consder just one rreducble component of ts pullback to prncpal level p 2015 at the resdue characterstc p, whch we denote by Y. We know essentally nothng about Y, but we can show that t s well postoned, that the (generally nfnte-dmensonal) coherent cohomology of ts partal torodal compactfcaton Y tor stll satsfes Koecher s prncple, and that the ntersecton complex of Ytor s well postoned and satsfes the generalzaton of Morel s formula! We shall follow [34, Notaton and Conventons] unless otherwse specfed. Whle for practcal reasons we cannot explan everythng we need from the varous constructons of torodal and mnmal compactfcatons we need, we recommend the reader to make use of the reasonably detaled ndex and table of contents there, when lookng for the numerous defntons. For references to [34], the reader should also consult the errata avalable on the author s webste for correctons to some known errors and mprecsons. We wll sometmes use materals n [41] wthout thoroughly revewng them. 2. Well-postoned subsets and subschemes of good ntegral models 2.1. Background settng. Let p > 0 be a ratonal prme number. Let us repeat [41, Ass. 2.1] as follows: Assumpton Let X S be a scheme over the spectrum of a dscrete valuaton rng R 0 of mxed characterstcs (0, p), whch s the pullback of one the followng ntegral models n the lterature: (The varous notatons S 0, S 0, etc below are those n the works we cted, whch we wll freely use, but mostly only n proofs.) (Sm) A smooth ntegral model M S 0 = Spec(O F0,( )) defned as a modul of abelan schemes wth PEL structures at a neat level G(Ẑ ), as n [34, Ch. 1, 2, and 7], wth p and = G( q ). (When = {p}, t s shown n [34, Prop ] that the defnton n [34, Sec ] by somorphsm classes agrees wth the one n [34, Sec ] by (p) -sogeny classes, whch was Kottwtz s defnton n [30, Sec. 5].) (Nm) A flat ntegral model M S 0 = Spec(O F0,(p)) of a modul M S 0 = Spec(F 0 ) at a neat level G(Ẑ) (essentally the same as above, but q

5 COMPACTIFICATIONS OF SUBSCEMES 5 wth = ) defned by takng normalzatons over certan auxlary good reducton models as n [36, Sec. 6] (whch allow bad reductons due to arbtrarly hgh levels, ramfcatons, and collectons of sogenes). (In ths case, we also allow F 0 to be a fnte extenson of the reflex feld, wth M etc replaced wth ther pullbacks.) For smplcty, we shall assume that, n the choce of the collecton {(L j,, j )} j J n [36, Sec. 2], we have (L j0,, j0 ) = (p r0 L, p 2r0, ) for some j 0 J and some r 0. (Spl) A flat ntegral model M spl Spec(O K) of M F 0 K Spec(K) defned by takng normalzatons as n [38, Sec. 2.4] over the splttng models defned by Pappas Rapoport as n [62, Sec. 15]. (By takng normalzatons, we mean we also allow to be arbtrarly hgher levels, not just the same levels consdered n [62, Sec. 15].) For smplcty, we shall assume that, n the choce of the collecton {(L j,, j )} j J n [38, Choces 2.2.9], we have (L j0,, j0 ) = (p r0 L, p 2r0, ) for some j 0 J and some r 0. (dg) A flat ntegral model S K Spec(O E,(v) ) n the notaton of [44, Introducton] at a neat level K. For consstency wth the notaton n other cases, we shall denote K, E, and S K as, F 0, and M, respectvely, n what follows. Essentally by constructon, there exsts some auxlary good reducton Segel modul M aux Spec( (p) ) n Case (Sm) above, wth a fnte morphsm M M aux O F0,(v) extendng a closed mmerson (p) M M aux F 0. In all cases, there s some group functor G over Spec(), and some reflex feld F 0. In Cases (Sm), (Nm), and (Spl), the ntegral models are defned by (among other data) an ntegral PEL datum (O,, L,,, h 0 ) (cf. [34, Def ]), whch defnes the group functor G as n [34, Def ], and the reflex feld F 0 as n [34, Def ]. For techncal reasons, we shall nsst that [34, Cond ] s satsfed. In Cases (Nm) and (Spl), we allow the level to be arbtrarly hgh at p. In Case (dg), we stll have an ntegral PEL datum defnng the auxlary good reducton Segel modul M aux, whch we abusvely denote as (O,, L,,, h 0 ) (wth O =, wthout aux n the notaton), whch also defnes a group functor G aux wth an njectve homomorphsm G G aux. We shall say that we are n Case (Sm), (Nm), (Spl), or (dg) dependng on the case n Assumpton from where X S s pulled back. For each X S as n Assumpton 2.1.1, we have good torodal and mnmal compactfcatons X tor, S and Xmn S, whose qualtatve propertes we summarzed axomatcally n [41, Prop. 2.2], based on the constructons n [34], [36], [40], [38], and [44], whch we also repeat as follows, for the sake of clarty: Proposton Let X S be as above. Then there s a mnmal compactfcaton J X mn : X X mn over S, together wth a collecton of torodal compactfcatons J X tor : X X tor,, over S, labeled by certan compatble collectons of cone decompostons, satsfyng the followng propertes: (1) The structural morphsm X mn S s proper. For each, there s a proper surjectve structural morphsm, : Xtor, Xmn, whch s compatble wth J X mn and J X tor n the sense that J, X mn =, J X tor.,

6 6 KAI-WEN LAN AND BENOÎT STRO (2) X mn admts a stratfcaton by locally closed subschemes flat over S, each of whch s somorphc to an analogue of X (n Cases (Sm), (Nm), or (Spl)) or a fnte quotent of t (n Case (dg)). Moreover, the ncdence relaton among the strata s preserved under pullback to fbers. (3) Each s a set { } of cone decompostons wth the same ndex set as that of the strata of X mn, whch can be called the cusp labels for X. For smplcty, we shall suppress such cusp labels and denote the assocated objects wth the subscrpts gven by the strata. (4) For each stratum, the cone decomposton s a cone decomposton of some P, where P s the unon of the nteror P + of a homogenous selfadjont cone (see [3, Ch. 2]) and ts ratonal boundary components, whch s admssble wth respect to some arthmetc group Γ actng on P (and hence also on ). (For example, n the case of Segel modul, each P + can be dentfed wth the space of r r symmetrc postve defnte parngs for some nteger r, and P can be dentfed wth the space of r r symmetrc postve sem-defnte parngs wth ratonal radcals.) Then has a subset + formng a cone decomposton of P+. If τ s a cone n that s not n +, then there exsts a stratum of X mn whose closure n Xmn contans, and a cone τ n +, whose Γ -orbt s unquely determned by the Γ-orbt of τ (where Γ s the analogous arthmetc group actng on.) We may and we shall assume that s smooth and projectve, and that, for each and σ +, ts stablzer Γ σ n Γ s trval. (5) For each, the assocated X tor, admts a stratfcaton by locally closed subschemes [σ] flat over S, labeled by the strata of X mn and the orbts [σ] + /Γ. The stratfcatons of Xtor, and Xmn are compatble wth each other n a precse sense: The premage of a stratum of X mn s the (settheoretc) dsjont unon of the strata [σ] of X tor, wth [σ] + /Γ. If τ s a face of a representatve σ of [σ], whch s dentfed (as n (4) above) wth the Γ -orbt [τ ] of some cone τ n +, where s a stratum whose closure n X mn contans, then [σ] s contaned n the closure of [τ ]. Such an ncdence relaton among strata s preserved under pullback to fbers. (6) For each stratum of X mn, there s a proper surjectve morphsm C from a normal scheme whch s flat over S, together wth a morphsm Ξ C of schemes whch s a torsor under the pullback of a splt torus E wth some character group S over Spec(), so that we have Ξ = Spec OC ( Ψ(l)) l S for some nvertble sheaves Ψ(l). (Each Ψ(l) can be vewed as the subsheaf of (Ξ C) O Ξ on whch E acts va the character l S.) (7) For each σ, consder σ := {l S : l, y 0, y }, σ 0 := {l S : l, y > 0, y }, σ := {l S : l, y = 0, y } = σ /σ 0. Then we have the affne torodal embeddng Ξ Ξ(σ) := Spec OC ( l σ Ψ(l)).

7 COMPACTIFICATIONS OF SUBSCEMES 7 The scheme Ξ(σ) has a closed subscheme Ξ σ defned by the deal sheaf correspondng to Ψ(l), so that Ξ σ = SpecOC ( Ψ(l)). Then Ξ(σ) l σ0 l σ admts a natural stratfcaton by Ξ τ, where τ are the faces of σ n. (8) For each representatve σ + of an orbt [σ] + /Γ, let X σ denote the formal completon of Ξ(σ) along Ξ σ, and let (X tor, ) [σ] denote the formal completon of X tor, along [σ]. Then there s a canoncal somorphsm X σ = (X tor, ) [σ] nducng a canoncal somorphsm Ξ σ = [σ]. (9) Let x be a pont of Ξ σ, whch can be canoncally dentfed wth a pont of [σ] va the above somorphsm. Let us equp Ξ(σ) wth a coarser stratfcaton nduced by the Γ-orbts [τ] of τ, where τ are the faces of σ. Each such orbt [τ] can be dentfed wth the Γ -orbts [τ ] of some cone τ n +, where s a stratum whose closure n X mn contans. Then there exsts an étale neghborhood U X tor, of x and an étale morphsm U Ξ(σ) such that the stratfcaton of U nduced by that of X tor, concdes wth the stratfcaton of U nduced by that of Ξ(σ), n the sense that the premage of the stratum [τ ] of X tor, concdes wth the premage of the [τ]-stratum of Ξ(σ) when [τ] determnes [τ ] as explaned above; and such that the pullbacks of these étale morphsms to [σ] and to Ξ σ are both open mmersons. (In partcular, X tor, and Ξ(σ), equpped wth ther stratfcatons as explaned above, are étale locally somorphc at x.) For our purpose n ths artcle, t s useful to have the followng more precse verson of (8) of Proposton 2.1.2: Proposton Let us retan the same settng as n Proposton For each gven, and for each, consder the full torodal embeddng Ξ = Ξ(σ) σ defned by the cone decomposton (cf. [34, Thm and Sec ]), and consder the formal completon X of Ξ along ts closed subscheme Ξ τ. τ + Consder, for each σ +, the formal completon X σ of Ξ(σ) along ts closed subscheme Ξ(σ) + := Ξ τ. Then X admts an open coverng by X τ +, τ σ σ for σ runnng through elements of +, and we have canoncal flat morphsms (2.1.4) X σ X σ X X tor, nducng somorphsms (2.1.5) X σ (X tor,) and (2.1.6) X /Γ (X tor,) τ +, τ σ [τ] [τ] [τ] + /Γ such that (2.1.6) nduces (2.1.5) by restrcton, extendng the X σ (X tor, ) [σ] and Ξ σ [σ] n (8) of Proposton More precsely, for each σ +, and for each affne open formal subscheme W = Spf(R) of X σ, under the canoncally nduced (flat) morphsms W := Spec(R) X tor, and Spec(R) Ξ(σ) nduced by (2.1.5), the stratfcaton of W nduced by that of X tor, concdes wth the stratfcaton of W nduced by that of Ξ(σ). In

8 8 KAI-WEN LAN AND BENOÎT STRO partcular, the premages of X and Ξ concde as open subschemes of W. Ths open subscheme, whch we denote as W 0, s the locus over Spec(R) where the pullback of any Mumford famly s abelan. (For the meanng of Mumford famles, see [34, Def ] n Case (Sm), and see [36, (8.29)] and [40, proof of Lem. 4.13] n Case (Nm). In Case (Spl), the Mumford famles are the pullbacks from those n Case (Nm); see [38, proof of Lem ]. In Case (dg), we consder any pullbacks of Mumford famles from auxlary torodal compactfcatons n Case (Sm).) Proof. In Case (Sm), these follow from the very constructon of Mumford famles as relatve schemes (wth addtonal structures) over the formal boundary charts n [34, Sec ], and from the proof of [34, Thm (5)] and ts modfcaton n the proof of [35, Lem ], based on [34, Thm (6)], by matchng the pullback = Mtor, wth the Mumford = X Φ,δ,, for each representatve (Φ Φ,δ, δ, σ) as n [34, Def ]. (Snce the pullback of the stratfcaton of X tor, s determned by the theory of degeneraton, t concdes wth the pullback of the stratfcaton of Ξ(σ); see the proofs of [34, Prop and ].) In Case (Nm), these follow from [40, Prop. 5.1 and 5.18]. In Case (Spl), these follow from the proposton n of the tautologcal sem-abelan scheme over X tor, famly over X Case (Nm), because the stratfcaton of X spl,tor, because Ξ spl Φ,δ (σ) = Ξ Φ,δ (σ) C Φ,δ C spl Φ,δ s the pullback of that of X tor,, and (see [38, (3.2.13)]). In Case (dg), ths follows from the facts that X tor, s the normalzaton (and hence fnte) over some auxlary good reducton torodal compactfcaton n Case (Sm), and that s open and closed n the premage of a stratum of ths each stratum [σ] of X tor, auxlary torodal compactfcaton, by the proof of [44, Prop ]. Let us also record the followng strengthenng of (9) of Proposton 2.1.2, whch follows from the same argument of the proofs of [41, Prop. 2.2(9) and Cor. 2.4], but wth the nput [41, Prop. 2.2(8)] there replaced wth Proposton here: Corollary (cf. [41, Cor. 2.4]). Let x be any pont of X tor,, whch we may assume to le on some stratum [σ]. Let σ be any representatve of [σ], and let E E(σ) and E σ be the affne torodal embeddng and the closed σ-stratum of E(σ) over Spec() (defned analogously as n the case of Ξ Ξ(σ) and Ξ σ, but are smpler). Then there exsts an étale neghborhood U X tor, of x and an étale morphsm and by U E(σ) C such that the stratfcatons of U nduced by that of X tor, Spec() that of E(σ) concde wth each other; and such that the pullback of U X tor the canoncal morphsm X tor, X mn X tor, under the canoncal morphsm E(σ) + Spec() τ + :τ σ E τ E(σ), are both open mmersons., under and the pullback of U E(σ) C Spec() C, where E(σ) + := C E(σ) Spec() Suppose τ s a face of σ. Then the premage of the stratum [τ ] of X tor, n U, where [τ ] s determned by [τ] as n (9) of Proposton 2.1.2, s the premage of the stratum E τ of E(σ). If we denote by tor [τ ] the closure of [τ ] n X tor,, and by E τ (σ) the closure of E τ n E(σ), then the above mples that, étale locally at x, the open mmerson J tor : [τ ] [τ ] tor [τ ] can be dentfed wth the product of the canoncal open mmerson J Eτ (σ) : E τ E τ (σ) wth the dentfy morphsm on C.

9 COMPACTIFICATIONS OF SUBSCEMES 9 In partcular, when τ = {0}, ths means the premage U of X n U concdes wth the premage of E. Moreover, étale locally at x, the open mmerson J X tor :, X X tor, can be dentfed wth the product of the canoncal open mmerson J E(σ) : E E(σ) wth the dentty morphsm Id C on C. In the remander of ths subsecton, we record some specal cases where C s known to be an abelan scheme torsor for each. Remark Already n Case (Sm), where G(Ẑ ) has no factor at p, the morphsm C mght not be an abelan scheme for each, for an arbtrary see the errata for [34] on the author s webste, and also the clarfcaton n [35, Rem ]. (It s only an abelan scheme torsor over a fnte étale cover of.) Nevertheless, C s ndeed an abelan scheme for all when s a prncpal level U (m) := ker(g(ẑ ) G(/m)) for some nteger prme m to, because the constructons n [34, Sec ] reman vald, despte the mstake when takng quotents n [34, Sec ]. Consder the followng specal case of Case (Nm) n Assumpton 2.1.1: Suppose p s a good prme (as n [34, Def ]) for an ntegral PEL datum (O,, L,,, h 0 ) as n Assumpton (whch we have nssted to satsfy [34, Cond ]). Consder the trval collecton J = {j 0 } wth {(g j0, L j0,, j0 } = {(1, L,, }, as n [36, Ex. 2.3]. Let be the prncpal level U(n) := ker(g(ẑ) G(/n)) for some n = n 0p r, where n 0 3 s an nteger prme to p, and where r 0 1, so that p = U p (n 0 ) := ker(g(ẑp ) G(/n 0 )) s neat. Let 0 := U(n 0 ) := ker(g(ẑ) G(/n 0)) = p G( p ). Then X S (resp. X 0 S) s a pullback of M S 0 (resp. M0 S 0 ) under some morphsm S S 0 = Spec(O F0,(p)). Lemma Wth the settng as above n Case (Nm), the morphsm C at level s an abelan scheme for all. Moreover, f we denote by C 0 0 the analogous morphsm at level 0, then the canoncal morphsm C C 0 can be 0 dentfed wth the multplcaton by p r on the abelan scheme C over. Proof. Let (, Φ, δ ) be a representatve of cusp label for M. Let ( p, Φ p, δ p) denote the prme-to-p part of (, Φ, δ ), and let ( 0, Φ 0, δ 0 ) denote the nduced representatve at level 0. It suffces to show that, n the notaton of [36, Sec. 8], CΦ,δ M s an abelan scheme, and the canoncal morphsm C Φ,δ C Φ0,δ 0 M can be dentfed wth the multplcaton M 0 0 by p r on the abelan scheme C Φ,δ over M. Snce p s a good prme for (O,, L,,, h 0 ), we have C Φ p,δ p M p p Spec(O F0,(p)) as n [34, Sec ], where the frst morphsm s an abelan scheme. By [34, Prop ], the canoncal morphsm M 0 0 M p p s an open and closed mmerson. Snce the schemes M 0 0 and M over S 0 = Spec(O F0,(p)) are ndependent of the auxlary choces (see [36, Prop. 6.1 and 7.4]), by takng M p p mmerson M 0 0 as an auxlary good reducton model, we have an open and closed M p, and we can take M p under the composton M M 0 0 M 0 0 to be the normalzaton of M p p of canoncal morphsms. M p p

10 10 KAI-WEN LAN AND BENOÎT STRO M 0 0 M Snce p s a prncpal level, by the constructon n [34, Sec ], C Φ p,δ p M p s an abelan scheme. By the same reasonng as n the prevous paragraph, p C Φ0,δ 0 M 0 0 s canoncally somorphc to the pullback of C Φ p,δ p M p p under the open and closed mmerson M 0 0 M p, whch s also an abelan p scheme. Snce = U(n 0 p r ) and 0 = U(n 0 ) are both prncpal levels, the canoncal morphsm C Φ,δ C Φ0,δ 0 can be dentfed wth the multplcaton by p r on the abelan scheme C Φ,δ over M. ence, there s also an somorphsm C Φ,δ CΦ0,δ 0 M, whch extends to an somorphsm M 0 0 C Φ,δ CΦ0,δ 0 M 0 0 M, by arsk s man theorem, because abelan schemes are smooth, and because the base scheme M s noetheran normal by constructon. The assertons n the frst paragraph above now follow. Lemma In Case (Spl) that s based on the settng as above n Case (Nm), the analogous statements n Lemma also hold. Proof. For each representatve (, Φ, δ ) of cusp label for M, by [38, Def ], C spl Φ,δ s the normalzaton of C Φ,δ M,spl M. When C Φ,δ M s an abelan scheme, ths fber product s already normal because M,spl s. ence, M,spl s the pullback of C Φ,δ M, and the lemma follows. C spl Φ,δ Remark Other than the above specal cases n Cases (Nm) and (Spl) (wth the restrctve assumpton that J = {j 0 }), t s also true n many other specal cases that C s an abelan scheme, or at least an abelan scheme torsor, for each. See, for example, the Segel modul wth parahorc levels n [71]. (It s plausble that the argument there can be generalzes to all cases n Cases (Nm) and (Spl) where p s good for (O,, L,,, h 0 ), and where = p p for some prncpal p G(Ẑp ) and for some parahorc subgroup p of G( p ) that s the dentty component of the stablzer of the base change of the collecton L to p.) 2.2. Well-postoned subsets and subschemes. Let T be a locally noetheran scheme over S. For the schemes X etc (and morphsms among them) over S, we shall denote ther pullbacks to T by (X ) T etc. Defnton We say that a locally closed subset (resp. subscheme) Y of (X ) T s well-postoned f there exsts a collecton Y = {Y } ndexed by the strata of X mn, where each Y s a locally closed subset (resp. subscheme) of T such that, for each W as n Proposton 2.1.3, the pullback of Y to (W 0 ) T under the nduced morphsm W 0 X s, as a subset (resp. subscheme) of (W 0 ) T, the pullback of Y under the composton W 0 of the nduced morphsm W 0 Ξ wth the canoncal morphsms Ξ C. For convenence, for each Y as above, we shall also denote by Y C the pullback of Y under C, as a subset (resp. subscheme). We shall say that Y s assocated wth Y.

11 COMPACTIFICATIONS OF SUBSCEMES 11 Lemma In Defnton 2.2.1, t suffces to verfy the condton for some affne open coverng of X = X σ + σ by affne formal schemes W s as n Proposton for just one collecton of cone decompostons. Proof. Snce locally closed subsets or subschemes concde f and only f they do over the open subsets n an open coverng, for each, t suffces to verfy the condton n Defnton for some open coverng as n the statement of the lemma. Snce every two s has a common refnement, t suffces to show that, f s a refnement of, then the condton holds for f and only f t holds for. In ths case, the canoncal morphsm X tor, Xtor, reduced subscheme of the premage of [σ], whch nduces a proper mor- [σ] + /Γ phsm (X tor, ) [σ ],+ [σ ] /Γ (X tor, ) s proper, under whch [σ ],+ /Γ [σ ] s the [σ] [σ] + /Γ On the other hand, for each σ +, consder Ξ(σ) := between the formal completons. τ,+, τ σ Ξ(τ), whch τ,+ X τ, τ σ admts a proper morphsm Ξ(σ) Ξ(σ) extendng the dentfy morphsm on Ξ and nducng a proper morphsm X σ compatble wth the composton of morphsms X τ X /Γ (X tor, ) (X tor, ) [σ ],+ [σ ] /Γ and X σ X /Γ [σ], where the frst morphsms are open mmersons, and where the [σ] + /Γ second morphsms are the canoncal somorphsms, as n Proposton Therefore, for each affne open formal subscheme W = Spf(R) of X σ, whch nduces a canoncal morphsm W = Spec(R) Ξ(σ), the pullback of W under Ξ(σ) Ξ(σ) s covered by fntely many W = Spec(R ), where W Ξ(σ) s nduced by some affne open formal subscheme W = Spf(R ) of X τ, for some τ,+ such that τ σ. Snce Ξ(σ) Ξ(σ) nduces the dentty morphsm on Ξ by pullback on the target, W 0 = W 0 s an open coverng, and the lemma follows, as desred. The followng three lemmas follow mmedately from the defnton: Lemma Intersectons of well-postoned subsets (resp. subschemes) of (X ) T are well-postoned subsets (resp. subschemes). Lemma Suppose Y s a well-postoned subset (resp. subscheme) of (X ) T,. If Y 0 s a closed subset of Y that s a well-postoned subset of (X ) T, then the open complement Y Y 0 s a well-postoned subset (resp. subscheme). Smlarly, f Y 0 s an open subset of Y that s a well-postoned subset of (X ) T, then the closed complement Y Y 0 s a well-postoned subset of (X ) T. Lemma Suppose a locally closed subset (resp. subscheme) Y of (X ) T s a unon of ts open subsets (resp. subschemes) {Y } I, where each Y s a wellpostoned subset (resp. subscheme) of (X ) T. In the case of subschemes, suppose moreover that, for each, the assocated subschemes {Y, } I nduce compatble open subscheme structures over ther fnte ntersectons, so that I Y, s defned. Then Y s also a well-postoned subset (resp. subscheme).

12 12 KAI-WEN LAN AND BENOÎT STRO Lemma Suppose Y s a closed well-postoned subset (resp. subscheme) of (X ) T, and suppose Y = {Y } s assocated wth Y as n Defnton Then Y s closed n T, and Y C s closed n C T, for each. Proof. For each, snce C s proper and surjectve, t suffces to show that Y C s closed n C T. Assume otherwse, amng for a contradcton. Then there exsts a pont x n the closure of Y C n C T, but not n Y C. Snce Ξ(σ) C and Ξ σ C are fathfully flat, by [21, IV-2, ], there exsts a pont y n the fber of (Ξ σ ) T C T above x such that, for every affne neghborhood U of y n Ξ(σ) T, the pullback of Y C to U 0 := U Ξ T s not closed. Snce X σ = (Ξ(σ)) Ξ σ Ξ(σ) s flat and nduces the dentty morphsm along Ξ σ, for some W as n Proposton 2.1.3, whch we may assume to contan y, the pullback Y W 0 of Y C to (W 0 ) T s not closed. But ths contradcts the assumpton that Y s closed, because Y W 0 s by defnton also the pullback of Y to (W 0 ) T, as desred. Lemma Suppose Y s a well-postoned subset (resp. subscheme) of (X ) T, and suppose Y = {Y } s assocated wth Y as n Defnton (Under the assumpton that T s locally noetheran, the morphsms Y (X ) T and Y are automatcally quas-compact, for all.) Then the closure (resp. schematc closure; see [9, Sec. 2.5, p. 55]) Y of Y n (X ) T s a well-postoned subset (resp. subscheme) f and only f the followng condton holds: For each, the closure (resp. schematc closure) Y C of Y C n C s the pullback of the closure (resp. schematc closure) Y of Y n, as a subset (resp. subscheme). In ths case, the closed complement Y 0 := Y Y s a well-postoned subset. (The above condton s automatcally satsfed when Y s closed n (X ) T, by Lemma 2.2.6; or when C s flat for each, by [21, IV-2, ] and [9, Sec. 2.5, Prop. 2].) In ths case, Y := {Y } (resp. Y 0 := {Y Y } ) s assocated wth Y (resp. Y 0 ) as n Defnton Proof. For each W as n Proposton 2.1.3, let Y W 0 denote the pullback of Y to W 0, as n the proof of Lemma Snce W 0 X s flat, by [21, IV-2, ] (resp. [9, Sec. 2.5, Prop. 2]), the closure (resp. schematc closure) Y W 0 of Y W 0 n (W 0 ) T concdes wth the pullback of Y. Smlarly, snce W 0 Ξ and Ξ C are flat, Y W 0 concdes wth the pullback of Y C. ence, by defnton, Y s well postoned f and only f the condton n the lemma holds. Defnton Let {S } I be a fnte set of subschemes of a scheme S such that each S s a closed subset of the set-theoretc unon S. For each I, let I S denote the schematc closure of S n S, and let S,0 := S S denote the closed complement. Then we defne the unon S as a scheme as the (closed) schematc I mage of S S (see [9, Sec. 2.5, p. 55]) subtracted by the closed subset S,0. I Lemma Suppose a locally closed subset (resp. subscheme) Y of (X ) T s a fnte unon of ts closed subsets (resp. subschemes) {Y } I (see Defnton 2.2.8), where each Y s a well-postoned subset (resp. subscheme) of (X ) T. For each I, suppose Y = {Y, } s assocated wth Y as n Defnton In the case of subschemes, suppose moreover that C s flat, for each. Then Y s also a well-postoned subset (resp. subscheme), and Y := {Y }, where Y := I Y, as

13 COMPACTIFICATIONS OF SUBSCEMES 13 a subset (resp. subscheme) for each, s assocated wth Y as n Defnton (Implct n ths statement s that each Y s defned n the case of subschemes.) Proof. Suppose x 0 s any pont of Y 0, whch specalzes to a pont x 1 of Y 1,, for some 0, 1 I. For each σ +, snce C s proper and surjectve, and snce Ξ(σ) C and Ξ σ C are fathfully flat (see [36, Prop and ts proof]), there exst some W as n Proposton and some ponts x 0 and x 1 of W 0 lftng x 0 and x 1, respectvely, such that x 0 specalzes to x 1. But x 0 and x 1 belong to the pullbacks of Y 0 and Y 1, respectvely. Therefore, the assumpton that Y 0 s closed n Y shows that x 1 s contaned n the pullback of Y 0, and so ts mage x 1 s contaned n Y 0,. ence, Y s closed n the set-theoretc unon 0, Y := I Y,, and ths unon s locally closed as a subset of T. In the case of subsets, Y := {Y } s assocated wth Y as n Defnton In the case of subschemes, each Y also admts the structure as a subscheme of T by Defnton 2.2.8, and Y := {Y } s assocated wth Y as n Defnton by [9, Sec. 2.5, Prop. 2], because W 0 X and W 0 C are flat, and because C s flat by assumpton. Lemma If Y s a well-postoned subset (resp. subscheme) of (X ) T, f Y = {Y } s assocated wth Y as n Defnton 2.2.1, and f C s reduced (.e., s flat and has geometrcally reduced fbers; see [21, IV-2, 6.8.2]) for all, then the unque reduced subscheme Y red over the underlyng locally closed subset Y of (X ) T s a well-postoned subscheme, and Y red = {Y red, }, where Y red, := (Y ) red for each, s assocated wth Y red as n Defnton Moreover, f T = Spec(k) for some feld k, and f C T T s (proper and) smooth for all, then the smooth locus Y sm of Y red s a nce well-postoned subscheme, and Y sm = {Y sm, }, where Y sm, := (Y ) sm s the smooth locus of (Y ) red for each, s assocated wth Y sm as n Defnton Proof. For each W as n Proposton 2.1.3, by the regularty of W (X ) T and W Ξ(σ) (see [21, IV-2, 7.8.3(v)]), by the reducedness of C, and by [21, IV-2, 5.8.5, 6.4.1, and 6.5.3], the pullback of Y red to (W 0 ) T concdes wth the pullback of (Y ) red as reduced subschemes, because ther underlyng sets already concde. ence, Y red s a well-postoned subscheme. If T = Spec(k) and f C T T s smooth (whch s, n partcular, also regular), by [21, IV-2, 6.5.3] agan, the pullback of Y sm to (W 0 ) T concdes wth the pullback of the smooth locus (Y ) sm of (Y ) red. Snce Y sm s open n Y red (by [21, IV-2, ]), t s locally closed, and hence also a well-postoned subscheme Compactfcatons of well-postoned subschemes. Let T be a locally noetheran scheme over S, as n the begnnng of Secton 2.2. Defnton Suppose Y s a locally closed subset (resp. subscheme) of (X ) T. Let Y denote the closure (resp. schematc closure) of Y n (X ) T, and let Y 0 denote the complement Y Y. (In ths defnton, we do not assume that any of Y, Y, or Y 0 s well postoned.) Let Y mn and Y tor denote the closure (resp. schematc closure) of Y (or equvalently Y) n (X mn ) T and (X tor, ) T, respectvely; and let Y0 mn and Y0, tor denote the closure (resp. schematc closure) of Y 0 n (X mn ) T and (X tor, ) T, respectvely. In the case of subsets, we vew them as subschemes wth ther reduced

14 14 KAI-WEN LAN AND BENOÎT STRO subscheme structures. Let Y mn := Y mn Y0 mn and Y tor := Ytor Y0, tor, wth nduced open mmersons J Y mn : Y Y mn and J Y tor : Y Ytor over T. We shall call Y mn (resp. Y tor ) the partal mnmal (resp. torodal) compactfcaton of Y, wth the term partal suppressed when Y s a closed subscheme of (X ) T. Theorem (cf. [41, Prop. 2.2] or Proposton 2.1.2). For each well-postoned subset (resp. subscheme) Y of (X ) T wth a collecton Y = {Y } as n Defnton 2.2.1, ts partal mnmal and torodal compactfcatons J Y mn : Y Y mn and : Y Ytor as n Defnton satsfy the followng propertes: J Y tor (1) For each, the proper surjectve structural morphsm, : Xtor, Xmn nduces a proper surjectve structural morphsm Y, : Ytor Ymn (over T), so that J Y mn = Y, J Y tor. The structural morphsms Ymn T and T are projectve when Y s closed n (X ) T (under the assumpton Y tor n (4) of Proposton that s projectve). (2) Consder an ample nvertble sheaf ω X mn (2.3.3) Y tor Case (Sm), we take ω X mn over Xmn chosen as follows: In = ω as n [34, Thm (2)]. In Case (Nm), = ω M,J as n [36, Prop. 6.4]. In Case (Spl), we take ω X mn as n [38, Thm (3)]. In Case (dg), we take ω X mn to be any of the ample ω (k,µ) we take ω X mn = ωmn K M spl,mn,j as n [44, Thm (2)]. Then the pullback ω Y mn of ω X mn to Ymn s also ample, and ts further pullback ω Y tor sem-ample. When Y s closed n (X ) T, we have canoncal morphsms ( Proj k 0 Γ(Y tor ) (, ω k ) Proj Y tor k 0 to Y tor Γ(Y mn, ω k Y mn )) = Y mn, whch concdes wth the Sten factorzaton of Y, : Ytor Ymn. (3) The stratfcaton of X mn by locally closed subschemes nduces a stratfcaton of Y mn by locally closed subschemes Y := Y mn, each of whch s equpped wth a canoncal morphsm Y Y whch nduces a bjecton between the underlyng subsets of T (see Defnton 2.2.1), wth an open dense stratum Y = Y for = X. For each, the stratfcaton of X tor, by locally closed subschemes [σ] nduces a stratfcaton of Y tor by locally closed subschemes Y [σ] := [σ] Y X tor tor, wth an open dense stratum, Y [{0}] = Y for = X and σ = {0}. For each [σ], the canoncal surjectve morphsm [σ] nduces a surjectve morphsm Y [σ] Y whch factors through a (surjectve) morphsm Y [σ] Y (whch s the pullback of [σ] when Y s a well-postoned subscheme of (X ) T ). ence, Y [σ] s nonempty exactly when Y s, and exactly when Y s. (4) For each top-dmensonal cone σ n +, we have a canoncal somorphsm Y [σ] Y C, whch shows that Y C s determned by Y, for each. (5) For each representatve σ + of an orbt [σ] + /Γ, and for? = Ξ, Ξ(σ), Ξ σ, X σ, X σ, and X, let Y? denote the pullback of? under the canoncal morphsm Y C C. (In the case of subsets, we vew Y and Y C as subschemes of and C, respectvely, wth ther reduced subscheme structures, so that the above all make sense as statements for schemes and X mn s

15 COMPACTIFICATIONS OF SUBSCEMES 15 formal schemes. In ths case, C nduces a proper surjectve morphsm Y C Y of schemes, whch s the pullback of C when C s reduced;.e., s flat and has geometrcally reduced fbers, as n [21, IV-2, 6.8.2].) Then we have a canoncal somorphsm (2.3.4) Y X σ (Y tor ) Y [σ] nduced by the canoncal somorphsm X σ (X tor, ) [σ], extendng a canoncal somorphsm Y Ξ σ Y[σ] nduced by the canoncal somorphsm Ξ σ [σ] (see (8) of Proposton 2.1.2), whch extends to flat morphsms (2.3.5) Y X σ Y X σ Y X Y tor nduced by (2.1.4), nducng compatble canoncal somorphsms (2.3.6) Y X σ and (Y tor ) τ +, τ σ Y [τ] (2.3.7) Y X /Γ (Y tor ) [τ] + /Γ Y [τ] nduced by (2.1.5) and (2.1.6), respectvely. (6) For each σ +, and for each affne open formal subscheme Spf(R) of Y X σ, under the canoncally nduced (flat) morphsms Spec(R) Y tor and Spec(R) Y Ξ(σ) nduced by (2.3.4), the stratfcaton of Spec(R) nduced by that of Y tor concdes wth the stratfcaton of Spec(R) nduced by that of Y Ξ(σ). In partcular, the premages of Y and Y Ξ concde as open subschemes of Spec(R). Analogous statements are true for Y X and (2.3.6). σ (7) Let x be a pont of Y Ξ σ, whch can be canoncally dentfed wth a pont of Y [σ] va the above somorphsm. Then there exsts an étale neghborhood U Y tor of x and an étale morphsm U Y Ξ(σ) such that the stratfcaton of U nduced by that of Y tor concdes wth the stratfcaton of U nduced by that of Y Ξ(σ), or rather by that of Ξ(σ) as n (9) of Proposton 2.1.2; and such that the pullbacks of these étale morphsms to Y [σ] and to Y Ξ σ are both open mmersons. (In partcular, Y tor and Y Ξ(σ), equpped wth ther stratfcatons as explaned above, are étale locally somorphc at x.) There also exst étale morphsms as above wth the analogous but stronger property that ther pullbacks to Y [τ] and to Y τ +, τ σ τ +, τ σ Ξ τ are both open mmersons. Proof. Propertes (1) and (2), and the assertons concernng underlyng subsets n property (3), follow mmedately from the defntons. Let Y, Y mn, Y tor, Y 0, Y0 mn, and Y0, tor be as n Defnton For each, let Y C denote the closure (resp. schematc closure) of Y C n C T, let Y C,0 := Y C Y C denote the complement, let Y denote the schematc closure of Y n T, and let Y,0 := Y Y denote the complement. For each W as n Proposton 2.1.3, as n the proof of Lemma 2.2.7, by [21, IV-2, ], by the defnton of schematc

16 16 KAI-WEN LAN AND BENOÎT STRO closures, and by the flatness of W (X ) T and W Ξ(σ) C, the pullback of Y tor = Ytor Y0, tor to W T s the pullback of Y C = Y C Y C,0, as a subset (resp. subscheme). When Y s just gven as a subset of (X ) T, by the regularty of W (X ) T and W Ξ(σ) (see [21, IV-2, and 7.8.3(v)]), by the normalty of Ξ(σ) C (see [36, Prop. 8.14] and ts proof), and by [21, IV-2, 5.8.5, 5.8.6, 6.4.1, and 6.5.3], t follows that the pullback of Y tor pullback of Y C as reduced subschemes. to W T also concdes wth the By the defnton of W = Spec(R) by an affne open subscheme W = Spf(R) of X σ, by Proposton 2.1.3, the pullback of Y [σ] := [σ] under the canoncal X tor, Y tor somorphsm Ξ σ [σ] concdes wth the pullback of Y C, for all σ +, and hence all the assertons n propertes (5) and (6) follow. Snce the canoncal morphsm Ξ σ C s an somorphsm when σ s top dmensonal, ths shows that Y C s unquely determned by Y, and property (4) follows. Snce the canoncal morphsms Ξ σ C are surjectve, whose composton can be dentfed wth the surjecton [σ] nduced by, : Xtor, Xmn, the nduced morphsms Y [σ] = Y Ξ σ Y C Y Y are also surjectve, where the last morphsm nduces a bjecton between subsets of T, and hence all assertons n property (3) follow. Fnally, property (7) follows from propertes (5) and (6) by the same argument as n the proof of (9) n [41, Prop. 2.2], but wth the orgnal Artn s approxmaton used there (see [1, Thm. 1.12, and the proof of the corollares n Sec. 2]) replaced wth the nested approxmaton n [76, Thm. 2.9] and [68, Thm and 11.5], where the hypothess (n the notaton there) that A B 1 s noetheran s satsfed A 1 because Y Ξ σ Y[σ] s nduced by Ξ σ [σ], and because the stratfcatons of Y tor and Y Ξ(σ) are also nduced by those of Xtor, and Ξ(σ), respectvely. Remark In Case (Sm), the assertons for Y tor n Theorem shows that our noton of well-postoned subschemes s consstent wth the one ntroduced by Boxer s n [11, Sec. 3.4]. Remark Shmura subvaretes are not well postoned n general, even as subsets, and that s why ther compactfcatons are more dffcult to construct. Thanks to Theorem 2.3.2, we can slghtly weaken Defnton as follows: Lemma Suppose T s a locally noetheran scheme over S, and suppose Y s a locally subset of (X ) T such that, for each, there exsts some subset Y of T such that, for each W as n Proposton 2.1.3, the pullback of Y to (W 0 ) T concdes wth the pullback of Y. Then Y s automatcally locally closed n T. Proof. Let Y tor be as n Defnton Snce C s proper and surjectve, t suffces to show that the pullback Y C of Y s locally closed n C T. Snce W X tor, and W Ξ(σ) are flat, by [21, IV-2, ], the pullback of Y tor to W s also concdes wth the pullback of Y C. Snce W s arbtrary, we may and we shall assume that the σ nvolved s top dmensonal n +, n whch case the we have an somorphsm Y [σ] Y C nduced by [σ] C, as n (4) of Theorem Snce s locally closed n ( [σ] ) T by ts defnton as a pullback of Y, t follows that Y [σ] Y C s locally closed n C T, as desred.

17 COMPACTIFICATIONS OF SUBSCEMES 17 Thanks to (7) of Theorem 2.3.2, we also have the followng: Proposton If Y s a well-postoned subset (resp. subscheme) of (X ) T, and f Y C Y has connected fbers for all, then the open and closed subsets (resp. subschemes) of Y are well-postoned subsets (resp. subschemes), and ther closures n Y mn and Y tor are also open and closed n these partal compactfcatons. Proof. Suppose Y = {Y } s assocated wth Y as n Defnton Suppose Y 1 s an open and closed subset (resp. subscheme) of Y. Let Y 2 := Y Y 1 (wth ts open subscheme structure when Y s a subscheme). We clam that ther respectve closures Y1, tor and Ytor 2, n Ytor do not overlap. Suppose, to the contrary, that there exst some and σ + x Y [σ] Y tor 1, Ytor wth some pont 2,, whch we dentfy wth a pont of Y Ξ σ va the somorphsm Y Ξ σ = Y[σ] n (5) of Theorem Let U Y tor and U Y Ξ(σ) be étale morphsms as n (7) of Theorem 2.3.2, whose pullbacks to Y [σ] and to Y Ξ σ are both open mmersons. Up to replacng U wth an open subscheme, we may and we shall assume that U Y tor and U Y Ξ(σ) have connected fbers. Consder the open subscheme U := U Y tor Y of U, whch can be dentfed wth U Y Ξ(σ) Y Ξ because the étale morphsms match stratfcatons. Let y denote the mage of x n Y C. Snce U Y Ξ s étale, and snce Y Ξ Y Ξ(σ) s an affne torodal embeddng over Y C, whch s fberwse dense, for each, there exsts some pont x of the pullback of Y to U whch specalzes to x n U and s mapped to y n Y C. Snce Ξ C s a torus torsor, the fber of U Y Ξ Y C above y s connected, whch cannot overlap wth both the pullbacks of Y 1 and Y 2. ence, such an x cannot exst, and the clam follows. So Y1, tor s also open and closed n Ytor. Now consder Y,[σ] := Y [σ] Y, tor, for each. Snce Y C Y has connected fbers, so does the surjectve morphsm Y [σ] = Y Ξ σ Y. Consequently, the mage Y, of Y, [σ] n Y, whch necessarly concdes wth Y Y mn as a subset of T, s open and closed, and Y,[σ] concdes wth the pullback of Y,, for each, because Y 1, and Y 2, do not overlap ether. So Ymn 1 s also open and closed n Y mn. In Y [σ] (resp. Y ), let us equp Y, [σ] (resp. Y, ) wth ts reduced subscheme structure n the case of subsets, and wth the canoncal open subscheme structure n the case of subschemes. Then the canoncal somorphsm (2.1.5) nduces an somorphsm Y X Y σ Y, (Y, tor) Y,[τ]. Thus, for each W as n Proposton 2.1.3, the τ +, τ σ pullback of Y 1 to (W 0 ) T concdes wth the pullback of Y 1,. ence, Y 1 s well postoned, wth assocated Y 1 := {Y 1, } as n Defnton 2.2.1, as desred. Proposton If Y s a well-postoned subset of (X ) T, and f Y C Y s flat and has rreducble fbers for all, then the rreducble components of Y are well-postoned subsets, and ther closures n Y mn and Y tor are also rreducble components of these partal compactfcatons. Proof. Let Y 1 be an rreducble component of Y. By defnton, ts closures n Y mn and Y tor, respectvely, are rreducble components of these partal compactfcatons. It remans to show that Y 1 s well postoned. Let U Y tor and U Y Ξ(σ)

18 18 KAI-WEN LAN AND BENOÎT STRO be étale morphsms as n (7) of Theorem 2.3.2, whose pullbacks to Y [σ] and to Y Ξ σ are both open mmersons. By the constructon of U n the proof of (7) of Theorem 2.3.2, we may and we shall assume that t s a nested approxmaton of the pullback of Y to some affne formal scheme W = Spf(R) as n Proposton 2.1.3, wth assocated affne scheme W = Spec(R). Up to replacng U wth an open subscheme, we may and we shall assume that the étale morphsms U Y tor and U Y Ξ(σ) have rreducble fbers. By [21, IV-2, ], the pullback U 1 of Y 1 to U s ether empty or an rreducble component of U. Suppose U 1 s nonempty, wth generc pont η U 1, whch s maxmal among ponts of U 1. Snce the morphsms U Y Ξ(σ) Y C Y are flat and have rreducble fbers (by assumpton), by [21, IV-2, ] agan, the mage η of η U 1 n Y s maxmal among ponts of Y, whose closure {η } n Y s an rreducble component, and U 1 concdes wth the pullback of {η }. Snce Y C Y has rreducble and hence connected fbers, Y, : Ytor Ymn has connected fbers over Y. ence, over each connected component of Y, there s at most one rreducble component of the form {η } as above. Let Y 1, be the (dsjont) unon of such rreducble components. Then ts pullback to any W (assocated wth W) as above concdes wth the pullback of {η }, and hence wth the pullback of U 1, or rather of Y 1. Snce the affne formal schemes W as above form an open coverng of X, for each, t follows from Lemma that Y 1 s well postoned, as desred. By the same arguments as n the proofs of [36, Prop and 14.2, and Cor. 14.4], usng the regularty of W X and W Ξ(σ) (see [21, IV-2, and 7.8.3(v)]) for each W as n Proposton 2.1.3, and usng the facts that Ξ(σ) C s surjectve and smooth (under the assumpton n (4) of Proposton that s smooth), and that Ξ s fberwse dense n Ξ(σ) over C (see [36, Prop. 8.14] and ts proof for both facts), we obtan the followng for any well-postoned subset or subscheme Y of (X ) T as n Defnton 2.2.1, where we equpped Y wth the canoncal reduced subscheme structure when Y s only gven as a subset, wth partal torodal compactfcaton Y tor as n Defnton 2.3.1: Proposton (cf. [36, Prop. 14.1]). Under the assumpton (n (4) of Proposton 2.1.2) that s smooth, Y s reduced (resp. normal, resp. regular, resp. Cohen Macaulay, resp. (R ), resp. (S ), one property for each 0, resp. flat over T, resp. fathfully flat over T) f and only f Y tor s. Proposton (cf. [36, Prop. 14.2]). Let P be the property of beng one of the followng: reduced, geometrcally reduced, normal, geometrcally normal, regular, geometrcally regular, Cohen Macaulay, (R ), geometrc (R ), and (S ), one property for each 0 (see [21, IV-2, and 5.8.2]). Under the assumpton (n (4) of Proposton 2.1.2) that s smooth, the fber of Y tor T over some pont s of T satsfes property P f and only f the correspondng fber of the open subscheme Y T over s does. Corollary (cf. [36, Cor. 14.4]). Suppose that Y T has geometrcally normal fbers. Then all geometrc fbers of Y tor T have the same number of connected components, and the same s true for Y T. (The analogous statements are true f we consder rreducble components nstead of connected components.)