Limiting Distributions of Scaled Eigensections in a GIT-Setting

Transcription

1 Lmtg Dstrbutos of Scaled Egesectos a GIT-Settg Dssertato zur Erlagug des Doktorgrades der Naturwsseschafte a der Fakultät für Mathematk der Ruhr-Uverstät Bochum vorgelegt vo Dael Berger m Oktober 014

2 Cotets I Itroducto ad Statemet of Results 1 II Exstece of Tame Sequeces 6 III Uform Localzato Proposto 14 IV Fber Probablty Measure Sequece 18 IV.1 Defto of π : X Y ad Y IV. The Fber Probablty Measure Sequece Tame Case) IV.3 The Fber Probablty Measure Sequece No-Tame Case) V The k-fberg Π: X Ỹ 7 V.1 Costructo of the k-fberg Π: X Ỹ V. Fber Itegral Propertes of the k-fberg Π: X Ỹ VI Uform Covergece the Tame Case 38 VI.1 Uform Covergece of the Fber Probablty Measures VI. Uform Covergece of the Fber Dstrbuto Destes VII Uform Covergece the No-Tame Case 44 VII.1 Aalyss of π 1 y) VII. A Local Proposto Cocerg Fber Itegrato VII.3 Local Uform Covergece o R N0 Y VII.4 Global Uform Covergece o R N0 Y Idex of Notato 65 Refereces 68

3 I INTRODUCTION AND STATEMENT OF RESULTS I Itroducto ad Statemet of Results The am of ths thess s to exame the asymptotc geometry of a certa class of sequeces of egesectos of a le budle by descrbg the covergece propertes of a aturally assocated measure sequece. I ths dscusso, we let X be a projectve, ormal, purely m-dmesoal varety ad T = T C a complex torus wth the uque maxmal compact subgroup T = S 1) m. We equp X wth a algebrac acto φ:t X X of the complex torus T ad assume φ to be compatble wth the holomorphc structure of X. Furthermore, we wll fx a based pot free le budle p:l X over X ad a algebrac T-acto φ:t L L whch projects dow to φ so that the correspodg morphsms of the fbers L x are lear trasformatos. I the sequel, we wll refer to L as a T-learzed le budle. Moreover, let h be a T-varat smooth, hermta, postve budle metrc o L ad let t be the Le algebra of the compact form T T C. I ths cotext, there exsts cf. [Gu-St]) a aturally assocated momet map µ:x t wth respect to the Kähler form p ω = h gve by the formula p µ = 1 4 dc log h X where X deotes the fudametal vector feld of φ o the total space L the sequel we wll use the otato h = ). If Im µ), t s possble to defe a equvalece relato o the Zarsk ope, T- varat set of semstable pots X ss = { x X: cl T.x) µ 1 ) } gve by z 0 z 1 : cl T.z 0 ) cl T.z 1 ). Havg defed, t s possble cf.?[he-hu], pp ) to equp the duced quotet X ss / wth a uque, holomorphc structure of a complex space deoted by Xss /T so that the quotet map π :X ss X ss /T s holomorphc ad has the followg two characterstc propertes: 1. If O X deotes the sheaf of holomorphc fuctos o X ad O X ss /T ad s the sheaf of holomorphc fuctos o X ss /T, the π O X ss )T = O X ss /T.. We have cl T.x) cl T.y) f ad oly f π x) = π y). As a geeralzato of the work of Shffma, Tate ad Zeldtch cf. [S-T-Z]) ad the results of Huckleberry, Sebert cf. [Hu-Se]), we lk the geometry of the sequece of T-represetato spaces, gve by H 0 X, L ), as to the geometry of the quotet π :X ss Xss /T =: Y. To be more precse, we show that for each choce of Im µ), t s possble to costruct a coverget measure sequece whch localzes alog µ 1 ) by usg sequeces of T-egesectos s H 0 X, L ),.e. exp η).s = e π 1 η) s where η t, t Z 1

4 I INTRODUCTION AND STATEMENT OF RESULTS whose rescaled weghts asymptotcally approxmate the chose t approprately as. As a startg pot, we show cf. Theorem 1) that, gve Im µ), there exsts a fte cover { X } of Xss I cosstg of ope, π-saturated subsets ad a fte collecto { s ) } I, where s H 0 X, L ) of sequeces cosstg of T-egesectos such that the followg propertes are fulflled: 1. where O 1) for each.. X Xs ) := { x X : s x) 0 } for all bg eough. 3. X T.µ 1 ) = X T.µ 1 1 ) for all N0. The costructo of such a tame collecto wll be the frst step of the preset work ad volves the combatoral aalyss of the sets µ cl T.x)), x X ss, whch are kow to be covex polytopes t cf. [At]). The crucal step of the exstece proof s to cotrol the depedece of the geometry of µ cl T.x)) as x vares X ss. Eve the specal case X s = Xss,.e. where every π-fber s gve by a T-orbt, the shape ad posto of µ cl T.x)) ca geeral vary cosderably. After havg prove the exstece of { s ) } wth the aforemetoed propertes, t s possble to defe for each fte, ope cover U = {U } of Y subordate to { π X )},.e. U π X ), a fte collecto { } ν U = { ν } of sequeces of π-fber probablty measures o Zarsk ope subsets of X ss by usg the correspodg orm fuctos s wth respect to the hermta budle metrc h. The precse costructo of the collecto { } ν U s based o the observato that the ambguty of the orm sequece s ), whch s oly welldetermed up to scalar multplcato, ca be abolshed f oe cosders the ormalzed sequece gve by s s. Here s deotes the fber tegral of the fucto s over π y := π 1 y). The collecto { } ν U assocated to the tame collecto { s } s the gve by ν y) A) := A s s d [π y ] for A measurable ad y U Y 0 where A d [π y] s defed to be the tegral over A π 1 y) of the restrcto ω dm C π y π y wth a certa multplcty 1. Sce the dmeso k y = dm C π 1 y) of a fber π 1 y) for y Y ca chage as y moves Y ad sce the costructo of each ν y) volves k y, oe ca ot expect that ν U defes a uform object over the full base Y. However, t s possble to fd a Zarsk dese subset Y 0 Y, so that π 1 y) for y Y 0 are purely k-dmesoal complex varetes set X 0 := π 1 Y 0 )). Over ths set, t s reasoable to exame the covergece propertes of the measure sequece ν ) for each I. More precsely, f f C 0 X) s a cotuous fucto ad f f red deotes the reduced fucto o the base Y gve by the restrcto f µ 1 ) of the averaged fucto f x) = T f t.x) dν T wth respect to the Haar measure ν T, we prove the followg theorem. 1 All relevat detals of the theory of fber tegrato ca be foud [K].

5 I INTRODUCTION AND STATEMENT OF RESULTS Theorem 3. [Uform Covergece of the Tame Measure Sequece] For for every tame collecto { s } there exsts a fte cover U of Y wth U π X ) so that the collecto of fber probablty measures { } { } ν U assocated to s coverges uformly o Y 0 to the fber Drac measure of µ 1 ) X 0,.e. for every I ad every f C 0 X), we have y ) π 1 y) f dν y) f red uformly o U Y 0. Furthermore, f { } { } D U = D probablty destes gve by D y, ):t D y, t) := the followg covergece result ca be proved. deotes the correspodg collecto of cumulatve fber { } s s t π 1 y) d [π y ] for y U Y 0 Theorem 4. [Uform Covergece of the Tame Dstrbuto Sequece] For every t R ad every tame collecto { s } there exsts a fte cover U of Y wth U π X ) so that the collecto of cumulatve fber probablty destes { D U, t) } assocated to { s } coverges uformly o Y 0 to the zero fucto o Y 0,.e. for every I we have y D y, t) ) 0 uformly o U Y 0 π X ). I ths sese, we have show that each tame sequece of egesectos s ), attached to a prescrbed weght, gves rse to a sequece of fber measures over Y 0, whch depedetly of the choce of s ), localzes uformly alog the crtcal µ 1 ). The localzato property of the measure sequece ν U ) attached to the tame collecto { s )} s a cosequece of the fact that the correspodg sequeces of strctly plursubharmoc fuctos ϱ :X R gve by ϱ := 1 log s coverge alog wth all dervatves) uformly o compact sets to a strctly plursubharmoc fucto ϱ. It s crucal to ote that the restrcto ϱ π 1 y) of ϱ to each fber of the projecto π :X ss Y takes o ts T-varat mmum alog the uquely defed T-orbt T.x y π 1 y) gve by T.x y = µ 1 ) π 1 y). Usg ths observato, t s the possble to deduce estmates of the magtude of ϱ ad hece of e ϱ = s outsde a T-varat, relatvely compact tube of µ 1 ) as teds to fty cf. Theorem ). Apart form the determato of the asymptotc behavor of ϱ, whch plays a essetal role the proof of Theorem 3, 4, t s also ecessary to deal wth the followg ssue: The fact that ν U ) s defed over a o-compact base makes a drect applcato of the 3

6 I INTRODUCTION AND STATEMENT OF RESULTS stadard covergece theorems of measure theory cosderably more dffcult. Therefore, a good porto of the proof of the above covergece theorem wll be devoted to resolvg ths ssue by costructg a ew quotet Π: X Ỹ whch exteds the restrcted quotet π :X 0 Y 0, so that the followg dagram commutes X 0 X π X 0 Y 0 Ỹ Π ad so that all fbers of Π are compact ad of pure dmeso k. The costructo of ths equvarat, dmesoal-theoretcal flatteg Π: X Ỹ, whch s based o results of D. Barlet, allows us to realze the fber measure sequece ν U ) as a restrcto of a measure sequece defed o X. Sce X ad all ts fbers are compact, t s the possble to show the above metoed covergece propertes of { } ν U by applyg results cocerg the cotuty of fber tegrato. I the last part of ths work, we retur to the tal sequece of T-egesectos s ) ad exame the covergece propertes of the fber measure sequece duced by s s. Ulke the tame case, we frst have to face the problem that the fucto s s oly well defed for all y Y wth the property s π 1 y) 0. The task of defg a maxmal, -stable set Y, o whch we ca cosstetly wrte dow a measure sequece ν ) for all N bg eough attached to s ), aturally leads to the oto of a removable sgularty. Havg avalable ths cocept, whch s based o the dea that certa sgulartes ca be dvded out by multplyg s wth locally defed, varat holomorphc fuctos, we are able to uquely exted the measure sequece beyod ts orgal set of defto for all N 0 o a ope subset of Y 0 whch s gve by R N0 Y 0. Here, R N0 deotes the ope subset of all sgulartes y Y whch are removable for all N 0. Oce the maxmal set of defto gve by R N0 Y 0 s foud, we cotue our dscusso by aalyzg the covergece propertes of the measure sequece ν ) over Y 0 R N0 ad obta the followg two results. Theorem 5.b. [Uform Covergece of the Ital Dstrbuto Sequece] For fxed t R the sequece D, t)) coverges uformly o Y 0 R N0 to the zero fucto. Theorem 6.b. [Uform Covergece of the Ital Measure Sequece] For f C 0 X) the sequece y π 1 y) f dν y) coverges uformly over Y 0 R N0 to the reduced fucto f red. Sce the devato of the tal measure sequece ν ) duced by s ) from a tame, locally defed measure sequece ν ) s completely descrbed by the locally defed sequece of fuctos ) gve by s = s, t s ot surprsg that the proof of both 4 )

7 I INTRODUCTION AND STATEMENT OF RESULTS theorems s based o techcs we already used before whe provg Theorem 3, 4. These are combed wth certa aalytc facts about the growth propertes of as ϱ o π 1 y) ad as. Ackowledgemets. Frst of all the author would lke to express hs grattude to hs advsor Professor Ala Huckleberry who tated the project for ths thess ad provded permaet support ad advse wheever eeded from ts very begg. The umerous, depth dscussos cludg hs suggestos whch were dspesable for the progress of the PhD project ca ot be valued hgh eough ad had a sustaed mpact o the mathematcal educato of the author. Furthermore, the author s debted to the costructve remarks of Prof. Dr. Hezer who always had a ear for the author s matters ad who cotrbuted much of hs expertse ad tme especally the cotext of the theory of the momet map ad the Hlbert quotet. The umerous dscussos wth Prof. Dr. Wkelma ad Prof. Dr. Wurzbacher were also dspesable for the outcome of the preset PhD thess. Besdes, specal thaks are owed to Dr. Tsaov who geerously provde advce ad practcal help. Fally, the author would lke to thak the Studestftug des Deutsche Volkes for the facal support whch cluded a scholarshp for the author s years as a studet ad PhD caddate at the Ruhr-Uverstät Bochum ad for hs tme as a vstg studet at the Massachusetts Isttute of Techology. 5

8 II EXISTENCE OF TAME SEQUENCES II Exstece of Tame Sequeces The am of ths secto s to prove the followg theorem. Theorem 1. [Exstece of Tame Sequeces] If L X s as above ad Im µ), the we ca fd a fte cover { X } I of X ss cosstg of ope, π-saturated subsets ad a fte collecto { s ) } I, where s H 0 X, L ) of sequeces cosstg of T-egesectos such that the followg propertes are fulflled: 1. where O 1) for each.. X Xs ) for all bg eough. 3. X T.µ 1 ) = X T.µ 1 ) for all N 0. I the sequel, we wll refer to a collecto { s )} of sequeces of T-egesectos s H 0 X, L ) wth the aforemetoed propertes as tame. Before we proceed wth the proof of Theorem 1, let us cosder a example of a tame collecto. Example II.1. Let X = CP 1 CP 1 wth Käher form ω = p 1 ω FS + p ω FS where p : X CP 1 deotes the -th projecto, {0, 1}. Equp X wth the T = C -acto gve by t. [z 0 :z 1 ], [ζ 0 :ζ 1 ]) = [t z 0 :z 1 ], [t ζ 0 :ζ 1 ]) ad cosder the T-learzato o L = p 0 O CP 1) 1 p 1 O CP1 1) X duced by the T- acto o CP 4 gve by t.[u 0 :u 1 :u :u 3 :v] = [t u 0 :u 1 :u :t 1 u 3 :v]. Here we vew L embedded O CP 3 1) = CP 4 \ {[0:0:0:0:1]} realzed as the coe over X = { [u] CP 4 : u 0 u 3 u 1 u = 0, v = 0 } { [u] CP 4 : v = 0 } = CP 3. Furthermore, let H deote the stadard hermta metrc o O CP 3 1) whch s gve by H[u:v]) = u v for u = u 0, u 1, u, u 3 ) 0 ad whch s T-varat wth respect to the above acto. Defe h := H X. If = 0 ad µ deotes the momet map duced by the hermta budle metrc h, t follows that µ 1 0) = {[z 0 :z 1 ], [ζ 0 :ζ 1 ]): z 0 ζ 0 z 1 ζ 1 = 0} CP 1 CP 1. 6

9 II EXISTENCE OF TAME SEQUENCES A calculato shows that the oly egesectos s H 0 X, L ) whch ca be part of a tame sequeces the sese of Theorem 1 are gve by the collecto A tame collecto s for example gve by { } s,k = z k 0 z k 1ζ0 k ζ1 k, 0 k H 0 X, L ). {z 0ζ 1 ), z 1ζ 0 ) }. For the proof of the exstece of tame sequeces, we wll frst restate a mportat fact about the geometry of a arbtrary T-orbt closure cl T.x) where x X: Every mage µclt.x)) t of a arbtrary orbt closure clt.x) X s the covex hull of the mage of ftely may fxed pots x Fx T := {x X: t.x = x for all t T} cf. [At]),.e. µclt.x)) = Cov {µs x )} II.1) where S x = { σ x,jx cl T.x) Fx T, 1 j x m x } Fx T s a fte set ad Cov {µ S x )} deotes the covex hull of the correspodg mage µ S x ). Cosder the decomposto of the T-represetato space H 0 X, L) ts egespaces Cs k where 1 k m := dm C H 0 X, L). For ay o-empty J {1,..., m} let S J := {s j : j J} H 0 X, L) ad let S J t be the set of all characters correspodg to the set S J. We troduce the followg otato: For a o-empty J {1,..., m}, let M J := { x µ 1 ): s j x) 0 for all j J, s j x) = 0 for all j J }. Note that M J µ 1 ) s a T-varat, ope subset whch mght be empty for certa o-empty subdces J {1,..., m}. Furthermore, the collecto {M J } J s fte ad cover µ 1 ). The frst clam follows by the fact that dm C H 0 X, L) < ad the secod clam s a drect cosequece of the assumpto that L s base pot free. The ext lemma establshes a coecto betwee the geometry of the mage of T.x uder the momet map µ where x M J ad the covex set Cov S J. Lemma II.1. Let x M J, the t follows that ad, f x s ot a T-fxed pot, µclt.x)) = Cov S J µ T.x) = Relt Cov S J where Relt Cov S J deotes the relatve teror of S J t. Proof. Frst of all ote that sce the mage of cl T.x) s kow to be the covex hull of 7

10 II EXISTENCE OF TAME SEQUENCES the mage µ S x ) where S x are the T-fxed pots cl T.x), the frst clam follows as soo as we have show that µ σ x,jx ) S J for all σ x,jx S x. Let σ x,jx S x. Sce L s assumed to be base pot free there exsts at least oe T- egesecto s whch does ot vash at σ x,jx. As x M J, such a s s ecessarly gve by s = s j for j J. If j deotes ts correspodg character, we deduce µ σ x,jj ) = j whch s a drect cosequece of the followg reasog: Let D:A 0 L) A 1 L) be the uquely defed, hermta coecto assocated to h ad recall that we have the formula cf. [Gu-St]) D Xη s + 1µ η s = d dt t=0exp 1 t η).s. for all η t. II.) If we apply formula II. to s = s j at the fxed pot σ x,jx, we deduce µ η σ x,jx ) = j η) for all η t. So t follows µσ x,jx ) = j ad hece µ cl T.x)) = Cov S J as clamed. The secod clam follows from the fact that = µ x) for x M J µ 1 ) ad the geeral fact that cl f A)) = f cl A)) for a cotuous map f :X Y where X s compact ad A X. Q. E. D. For the proof of the ext proposto, whch wll be crucal for the exstece of tame sequeces, we eed the followg techcal lemma. Lemma II.. Let P = Cov {q 1,..., q m )}, q 1,, q m R ad let P 1 N 0 := Cov 1 N 0 {q 1,..., q m } := { p = j ν j q j : j ν j = 1, ν j 1 N 0 }. If P, the there exsts a sequece = j ν j,q j so that the followg codtos are fulflled: 1.. O 1) 3. P 1 N0 for all bg eough. 4. There exsts N 0 N ad a partto J = J 0 J 1, J 1 of J = {1,..., m} so that ν j, = 0 for all j J 0, N 0 ad ν j, > 0 for all j J 1, N The sequeces ν j, ) are coverget so that the lmts are strctly postve for all j J 1. Proof. Let P 0 R m deote the covex hull of the stadard bass {e 1,..., e m } R m, the P := Cov {q 1,..., q m } = MP 0 ) II.3) 8

11 II EXISTENCE OF TAME SEQUENCES for the matrx M whose colums are gve by q 1,..., q m ). By II.3 we fd ν P 0 wth = M ν). Wthout restrcto of geeralty, we ca assume that ν Relt Cov {e l,..., e m }, II.4) where 1 l m 1 because otherwse, = q j0 for oe 1 j 0 m ad the clam follows mmedately by choosg ν j0, = 1 ad ν j, = 0 for all j j 0. Now, t s drect to see that we ca choose a sequece ν ) Cov 1 N {e l,..., e 0 m } so that ν ν where all lmts are strctly postve. Furthermore, we ca always guaratee that ν ν O 1). Set ) = Mν )) P 1 N 0. The frst clam follows by = M ν ) M ν) =, the secod clam s a drect cosequece of M ν ν O 1) ad the thrd, resp. ffth clam follows by costructo. The fourth clam results from equato II.4: We have ν j, = 0 for all j wth 1 j l 1 ad ν j, > 0 for all j wth l j m ad bg eough whch follows by ν ν Relt Cov {e l,..., e m }. Hece, we have J 0 = {1,..., l 1} ad J 1 = {l,..., m}. Q. E. D. The followg proposto wll be the essetal step order to prove Theorem 1. Proposto II.1. If L X s as above ad Imµ), the for each x M J there exsts a sequece s J ), sj H 0 X, L ) of J -egesectos wth the followg characterstc propertes: 1. J where J O 1). The set Xs J ) := {x X:s J x) 0} s depedet of for bg eough. 3. X J := π 1 π M J )) Xs J ). 4. X J T.µ 1 ) = X J T.µ 1 J ) for all bg eough. Proof. By Lemma II.1 we kow that µ cl T.x)) = Cov S J. By applyg Lemma II. we fd a sequece ) J, Cov S J so that Card J J, = j=1 ν J j, j where ν J j, N {0} for all N ad ) coverges to so that J, O 1). 9

12 II EXISTENCE OF TAME SEQUENCES Note that we have CardJ j=1 νj, J = ad that Card J J, = νj, J j = νj, J j j J 1 j=1 recall that, accordg to Lemma II., we have a partto J = J 0 J 1, J 0, where ν J j, = 0 for j J 0). Cosder Card J s J := j=1 s νj j, j H 0 X, L CardJ j=1 ν J j, ) = H 0 X, L ), whch defes a sequece of egesectos whose assocated weght vectors J approxmate : J Card J = νj, J j. j=1 Furthermore, by the fourth clam of Lemma II., we kow that ν j, = 0 for all j J 0 ad ν j, > 0 for all j J 1 for all bg eough. Therefore we deduce X s J ) = {s j = 0} j J 1 for all bg eough whch proves the secod clam. The thrd property ca be proved as follows: If z π 1 π M J )), the we have cl T.z) T.M J. Sce cl T.z) s T-varat, t follows that T.M J cl T.z). As s J T.M J 0 for all bg eough, t follows that s J cl T.z) 0 ad hece s J z) 0 for all bg eough. It remas to verfy the fourth clam. For ths, let x X J T.µ 1 ). Note that we ca assume that x s ot a T fxed pot: If x s a T fxed pot, the t follows x M J µ 1 ) ad hece j = for all j J. I partcular, we fd = 1 J for all bg eough, so the fourth clam s mmedate. I the sequel, let x X J T.µ 1 ) be ot a T fxed pot. Observe that x T.M J ad hece µ T.x) or by Lemma II.1. Sce 1 J, Relt µ cl T.x)) = Relt Cov S J 1 J, ad Relt Cov S J, we have Relt µ cl T.x)) = Relt Cov S J for all bg eough as well. Therefore we deduce x T.µ 1 1 J, ) for all bg eough, whch proves the cluso. 10

13 II EXISTENCE OF TAME SEQUENCES Now, let x X J T.µ 1 1 J ). Note that, as the prevous case, we ca assume that x s ot a T-fxed pot. Moreover, let T.z x be the uque closed orbt π 1 π x)).e. π 1 π x)) T.µ 1 ) = T.z x. Note that z x T.M J. If the clam were false,.e. f T.x T.z x, the we would deduce Cov S J = µ cl T.z x )) bd µ cl T.x)) where 1 J, µ cl T.z x )) for all. I partcular, by the above cluso, t the follows that 1 J, / µ T.x) for all cotradcto to x XJ T.µ 1 1 ) J for all bg eough. Therefore, the assumpto s false ad t follows that x T.M J ad hece µ T.x) or x T.µ 1 ) whch proves. Q. E. D. After havg show Proposto II.1 we ca prove that the set of semstable pots X ss ca be covered by the -stable complemets of ftely may sequeces s ) of egesectos whose assocated weght sequeces ) approxmate the ray R 0. Theorem 1. [Exstece of Tame Sequeces] If L X s as above ad Im µ), the we ca fd a fte cover { X } I of X ss cosstg of ope, π-saturated subsets ad a fte collecto { s ) } I, where s H 0 X, L ) of sequeces cosstg of T-egesectos such that the followg propertes are fulflled: 1. where O 1) for each.. X Xs ) for all bg eough. 3. X T.µ 1 ) = X T.µ 1 ) for all N 0. Proof. Frst of all choose a dexg I of all o-empty M J ad recall that the collecto {M } I defes a fte cover µ 1 ). Hece, the correspodg collecto { X } of ope, I π-saturated subset X = π 1 π M )) defes a fte cover of X ss ad the clam s a drect cosequece of Proposto II.1. Q. E. D. Before we proceed wth the proof of Proposto II., we cosder the followg example of Theorem 1. Example II.. Let X = Σ m, m N, be the m-th Hrzebruch-Surface cf. [Hr]) whch s defed as the projectvzato P O CP 1 1) O CP 1 m)) ad whch s somorphc to the hypersurface {z m 0 ζ 1 z m 1 ζ = 0} CP 1 CP. 11

14 II EXISTENCE OF TAME SEQUENCES Cosder the T = C -acto o CP 5 gve by t.[u] = [u 0 :t u 1 :t u :u 3 :t u 4 :t u 5 ] whch pulls back to the C -acto o CP 1 CP gve by t. [z 0 :z 1 ], [ζ 0 :ζ 1 :ζ ]) = [z 0 :z 1 ], [ζ 0 :t ζ 1 :t ζ ]) va the Segre-embeddg σ 1, :CP 1 CP CP 5. Moreover, fx the C -learzato o O CP 5 1) CP 5 gve by t.[u, ζ] = [t.u:ζ] where we have used the detfcato O CP 5 1) = CP 6 \ {[0:...:0:1]}. It s drect to check that ths C -learzato ca be pulled back to a C -learzato o L := σ 1, O CP 5 1)) Σ m. Furthermore, the momet map of the C -learzato o O CP 5 1) CP 5 assocated to the stadard hermta metrc h o O CP 5 1) defed by h u 0, ζ 0 ), u 0, ζ 0 )) = u ζ 0 ζ 1 yelds a momet map µ:x t o X whch s gve by µ [z 0 :z 1 ], [ζ 0 :ζ 1 :ζ ]) = 1 z 0 ζ 1 + z 0 ζ + z 1 ζ 1 + z 1 ζ z 0 ζ 0 + z 0 ζ 1 + z 0 ζ + z 1 ζ 0 + z 1 ζ 1 + z 1 ζ. If = 1 Im µ = [0, 1 ], the µ 1 ) = ad { ) 1 z 0 ζ 1 + z 0 ζ + z 1 ζ 1 + z 1 ζ ) } z 0 ζ 0 z 1 ζ = 0 Σ m X ss = X \ {ζ 0 = 0 ζ 1 = ζ = 0}) CP 1 CP where π :X ss Y = CP 1 ca be detfed wth the restrcto p CP 1 Σ m = π of the projecto map p CP 1 :CP 1 CP CP 1. A further aalyss of the example shows that 6 = dm C H 0 Σ m, L) where s 1 = z 0 ζ 0, s = z 0 ζ 1, s 3 = z 0 ζ, s 4 = z 1 ζ 0, s 5 = z 1 ζ 1, s 6 = z 1 ζ 1 = 0, = 1, 3 = 1, 4 = 0, 5 = 1 6 = 1. Moreover, t s drect to check that M J f ad oly f J {{1, 3}, {4, 5}, J}. A tame collecto s for example gve by { X } =1, = { π 1 CP 1 \ {[1:0]} ), π 1 CP 1 \ {[0:1]} )} where { s ) }=1, = { s 1 s 3 ), s 4 s ) 5 }. We complete ths secto by provg the followg proposto. 1

15 II EXISTENCE OF TAME SEQUENCES Proposto II.. If s ) s a tame collecto, the the assocated collecto of sequeces of strctly plursubharmoc potetals ϱ ) gve by ϱ = 1 log s coverges uformly o every compact set of X to a smooth strctly plursubharmoc fucto ϱ :X R. Moreover, the same s true for all ts dervatves. Proof. Frst of all recall cf. proof of Proposto II.1) that each sequece s ) s gve by CardJ s J := s νj j, j, where J {1,..., m} s a fte, sutable dex set ad νj, J ) that Card J j=1 j=1 ν J j, j = j J 1 ν J j, j a sequece of tegers such where J = J 0 J 1, J 1, ν j = 0 for all j J 0 ad all bg eough, resp. ν j > 0 for all j J 1 ad all bg eough. Hece, t follows that ϱ = j J 1 ν J j,log s j, ν J j, > 0 for j J 1 for all bg eough. Recall that X s J ) = j J 1 {s j = 0} for all bg eough. As the sequeces νj, J ) are coverget wth strctly postve lmts for all j J 1, t follows that ϱ coverges uformly o compact subsets of X all dervatves to the smooth s.p.s.h. :=strctly plursubharmoc) fucto ϱ :X R gve by ϱ = j J 1 lm νj j, log s j. Q. E. D. 13

16 III UNIFORM LOCALIZATION PROPOSITION III Uform Localzato Proposto I ths secto fx a tame sequece s ) ad let ϱ ) be the assocated sequece of strctly plursubharmoc fuctos. Moreover, recall that by Theorem 1, the subset X s π-saturated ad cotaed the complemet of the zero set X s ) of s for all bg eough. We start ths secto by recallg the followg basc fact cf.?[he-hu], pp ). Lemma III.1. Let ϱ :X R 0 ad let Y = π X ) be as above, the for each y Y there exsts a π-saturated, ope subset V = π 1 W ) of X ss where W Y s ope so that ϱ π ) V s proper. Note that t s always possble to assume that W Y s a compact eghborhood whch we wll do form ow o. Moreover, sce Y s compact ftely may of those compact eghborhoods W wll cover Y. I the sequel, we wll work wth the ormalzed s.p.s.h fucto ˆϱ, resp. ˆϱ o X defed by ˆϱ := ϱ π ϱ red resp. ˆϱ := ϱ π ϱ,red. Sce π ϱ s cotuous ad sce W Y was chose to be a compact eghborhood, the above lemma remas vald for ˆϱ, resp. for ˆϱ. I the sequel, we wll set ϱ = ˆϱ ad ˆϱ = ϱ. We defe T ɛ, W ) := ϱ π ) 1 [0, ɛ] W ). By the above lemma, T ɛ, W ) s a relatvely compact subset of X X ss whch s T- varat by ts defto. Furthermore, defe T c ɛ, W ) := ϱ π ) 1 [ɛ, ) W ). Remark III.1. Note that there exsts N 0 ɛ) N so that µ 1 1 ) π 1 W ) T ɛ, W ) for all N 0 ɛ). Otherwse there would exst a sequece x ) µ 1 1 ) π 1 W ) so that x / Tɛ, W ) for all N bg eough. Sce X ad W are compact, we ca fd a coverget subsequece x j so that x )j j x 0 where π x 0 ) W ad x 0 µ 1 ) whch follows by x j µ 1 1 j ) ad j 1 j. Hece, we have x 0 µ 1 ) π 1 W ). O the other had, we have assumed that x j / T ɛ, W ) for all bg eough. However, sce T ɛ, W ) s a ope eghborhood of µ 1 ) π 1 W ) π 1 W ) ths yelds a cotradcto to x 0 µ 1 ) π 1 W ). Before we prove Theorem, we eed the followg preparato. Lemma III.. Let s ) be a tame sequece as above, π 1 y) X ad let T.z y be the uque closed orbt π 1 y) the t follows that ϱ π 1 y) T.z y takes o a uque mmum alog the set T.µ 1 1 ) π 1 y) 14

17 III UNIFORM LOCALIZATION PROPOSITION whch s cotaed T.µ 1 ) π 1 y). Proof. Note that sce s ) s a tame sequece we have T.µ 1 ) X = T.µ 1 1 ) X by the thrd clam of Theorem 1. Ths shows that the set T.µ 1 1 ) π 1 y) s cotaed T.µ 1 ) π 1 y) so the secod clam s proved. The frst clam s a drect cosequece of the fact that the uque mmum of ϱ o T.µ 1 1 ) π 1 y) s kow to be equal to T.µ 1 1 ) π 1 y). Q. E. D. We ca ow prove the uform Localzato Proposto. Theorem. [Uform Localzato of the Potetal Fuctos] Let s ) be a tame sequece ad ϱ ) the assocated sequece of strctly plursubharmoc fuctos. Let W π X ) ad ɛ > 0 be as above ad let δ > 0 be gve. The there exsts N 0 N so that ϱ x) ɛ δ for all x T c ɛ, W ) ad all N 0. Proof. Frst of all, by Remark III.1, we ca assume that µ 1 1 ) π 1 W ) T ɛ, W ) for all N 0 ɛ). III.1) Moreover, ote that ϱ coverges uformly o relatvely compact sets ad hece o T ɛ, W ). Therefore, we ca fd a N 0 N so that ϱ x) ɛ δ III.) for all x the compact subset ϱ π ) 1 {ɛ} W ) ad all N 0. We cotue the proof by cosderg the followg two cases: 1) Frst of all let, x T c ɛ, W ) T.z x where T.z x s the uque closed orbt the fber π 1 π x)). We have to show that ϱ T c ɛ, W ) T.z x > ɛ δ. Applyg Lemma III., we kow that the restrcto ϱ T.z x of the strctly plursubharmoc fucto ϱ o the uque closed orbt T.z x the fber π 1 π x)) X takes o 15

18 III UNIFORM LOCALIZATION PROPOSITION ts mmum alog µ 1 1 ) π 1 π x)) T.z x. If m x := A t x t deotes the ahlator {η t : η, = 0 for all t x } of the sotropy t x, the T.z x s somorphc to the homogeous vector budle wth typcal fber m x : T.z x = T T xz Note that the zero secto T Txz m z s mapped uder the above detfcato to the T-orbt T.µ 1 ) T.z x. Moreover, usg ths detfcato, t follows that the restrcto of ϱ o T.z x yelds a smooth fucto o T Txz m z whose restrcto o each fber [{t} m z ], t T s a strctly covex fucto wth a uque mmum gve by [{t} {m }] for m m z. Note that the tube T ɛ, W ) T.z x s somorphc to a tube of the zero secto T Tx 0 mx ad also ote that ths tube cotas [T {m }] for all bg eough by the remark at the begg of ths proof whch s based o Remark III.1. We cotue the proof by coectg x = [{t} {η x }], for η x m z sutable, ad the uque mmum m = [{t} {m }] of ϱ wth a le Λ:R [{t} m x ] the vector space [{t} m x ] so that Λ 0) = m ad Λ 1) = x. Note that ths le tersects ϱ, 1 ɛ), say Λ τ ) = y, where 0 < τ < x, because the mmum of ϱ s cotaed the tube T ɛ, W ) for all bg eough) whereas x s ot by our assumpto. To sum up, we have a covex fucto Λ ϱ o R wth a uque mmum at 0 so that Λ ϱ τ ) ɛ δ for all bg eough by equato III.. Hece, t follows that ϱ x) ɛ δ as well for all bg eough. ) The ext step s to show that the equalty ϱ x) ɛ δ also holds for all x T c ɛ, W ) so that x s ot cotaed the uque closed orbt T.z x of the fber π 1 π x)). Let us assume that ths s ot true. By the Hlbert Lemma cf. [Kra]), we ca fd a oe parameter group γ :C T so that mz lm γ t).x T.z x. t 0 Note that ether the pull back γ ϱ or the pull back γ ϱ attas ts S 1 -varat mmum o C sce otherwse t would follow that ϱ Im γ C ) ad ϱ Im γ C ) atta ther S 1 -varat mmum o T.x. However, ths would yeld a cotradcto to the assumpto T.x T.z x ad the clam of Lemma III.. Hece, t follows that t γ ϱ t) ad t γ ϱ t) where t C are strctly mootoe decreasg whe t 0 they ca ot be strctly creasg sce s does ot vash o T.z x ). The preset case ca be subdvded to the followg cases:.a) Assume that x 0 = lm t 0 γ t).x T.z x T c ɛ, W ), the we have ϱ x 0 ) ɛ δ 16

19 III UNIFORM LOCALIZATION PROPOSITION for all N 0 by case 1). Sce γ ϱ s mootoe decreasg for t 0 we deduce that for all N 0 as clamed..b) Assume that δ ɛ γ ϱ 1) = ϱ x) x 0 = lm t 0 γ t).x T.z x T ɛ, W ), the there exsts t C so that γ t).x ϱ, 1 ɛ) ad hece ϱ γ t).x) = ɛ δ for all N 0 by III.. As case.a), t the follows that ϱ x) δ ɛ for all N 0 as clamed. Q. E. D. We close ths secto wth the followg corollary whch slghtly geeralzes the clam of Theorem III. I the cotext of Theorem III, fx m 0 N ad cosder the sequece s s, 1 m 0 ) of meromorphc η,m0 := m 0 -egesectos. Sce we have s x) 0 for all x X ad all N ad hece partcular for = m 0, we ca defe ϱ,m0 := 1 log s s, 1 m 0. It s drect to check that the above defto yelds a sequece of smooth, strctly plursubharmoc, T-varat fuctos for all m 0 o the π-saturated ope set X. Furthermore, t s kow that ϱ,m0 π 1 y) for y π X ) takes o ts uque mmum alog the set µ 1 1 ) η,m0 π 1 y). Therefore, after havg appled the argumetato of Remark III.1, we ca assume, as at the begg of the proof of Theorem III, that µ 1 1 ) η,m0 π 1 W ) T ɛ, W ) for all bg eough. The proof of Theorem III ow traslates verbatm to the sequece ϱ m0 ) ad yelds the followg corollary. Corollary III.1. If δ > 0, m 0 N fxed, the t follows ϱ,m0 for all x T c ɛ, W ) ad all bg eough. x) ɛ δ 17

20 IV FIBER PROBABILITY MEASURE SEQUENCE IV Fber Probablty Measure Sequece IV.1 Defto of π : X Y ad Y 0 As before, let X be a purely m-dmesoal, ormal T-varety wth Kähler structure ω ad let Y = X ss /T be the assocated Hlbert quotet. Note that we ca always assume Y to be purely dmesoal,.e. = dm C Y: Sce X s ormal by our assumpto, t follows cf. [He-Hu1], p. 14) that Y s ormal as well. I partcular, t follows that Y s locally of pure dmeso cf. [Gr-Re], p. 15) ad by cosderg the coected compoets of Y, whch are fte umber, we ca cofe ourselves to the case where Y s of pure dmeso = dm C Y. Now, recall that X ss s Zarsk ope ad Zarsk dese X. Cosder the compact varety X defed by the ormalzato of cl Γ π, where Γ π deotes the graph of π,.e. Γ π := { x, y) X ss Y : π x) = y}. Furthermore, we defe the algebrac map π : X Y by π := p Y cl Γ π ζ, where ζ : X = cl Γ π ) or cl Γ π deotes the ormalzato map ad p Y the projecto map p Y :X Y Y. Moreover, edow X wth the T-acto duced by the lft for ts exstece see [Gr-Re], pp. 164 f.) of the T-acto o cl Γ π, defed by t. x, y) = t.x, y) ad equp X wth a smooth, )-form ω gve by ω := p X cl Γ π ) ζ) ω. For the sake of completeess, we ote the followg remark. Remark IV.1. The graph Γ π X Y s T-varat wth respect to the acto o X Y gve by t. x, y) = t.x, y). Sce Γ π s a Zarsk ope ad Zarsk dese subset of X, t follows that cl Γ π s T-varat. Moreover, as ζ s T-equvarat, all fbers π 1 y), y Y are T-varat as well. Note that sce X s a assumed to be of pure dmeso m, t follows that cl Γ π s lkewse a purely m-dmesoal subvarety of X Y. As ζ s fte, we deduce that X s of pure dmeso m too. The ext step s to fd a Zarsk ope subset Y 0 Y so that the fbers of the restrcted projecto π π 1 Y 0 ): π 1 Y 0 ) Y 0 are all purely k-dmesoal varetes. The exstece of Y 0 s a drect cosequece of kow facts complex aalyss ad algebrac geometry: By a theorem of Carta ad Remmert cf. [Loj], p. 71 f.), t follows that the subset E X defed by E := { x X } : k < dm C,x π 1 π x)) X s a proper aalytc subset of X where k = m = dm C X dmc Y. Hece, by Chow s Theorem t follows that E s a proper algebrac subset. Applyg the Drect Image Theorem cf. [Gr-Re], p. 07), oe deduces that the mage π E) s a proper aalytc subset of Y. I partcular t s a proper algebrac subvarety of Y. Now, set Y 0 := π E) ad ote that all fbers of π over Y 0 are purely k-dmesoal by costructo. 18

21 IV FIBER PROBABILITY MEASURE SEQUENCE For later use, we troduce the followg otato: Let X, Y be purely dmesoal complex spaces m = dm C X, = dm C Y) where Y s assumed to be ormal ad let F:X Y be a holomorphc map so that all o-empty fbers F 1 y) are of pure dmeso k = m. The F s called a k-fberg. Note that π π 1 Y 0 ) : π 1 Y 0 ) Y 0 s a k- fberg. A example for the costructo of X as descrbed above s gve by the ext example. Example IV.1. Let X = CP 1 CP 1 wth the Käher form ω = p 1 ω FS + p ω FS where p : X CP 1 deotes the -th projecto, {0, 1} equpped wth the T = C -acto gve by t. [z 0 z 1 ], [ζ 0 :ζ 1 ]) = [t.z 0 :z 1 ], [t.ζ 0 :ζ 1 ]) ad cosder the momet map gve by cf. Example II.1) I partcular, we have X ss = 1 µ [z 0 :z 1 ], [ζ 0 :ζ 1 ]) = z 1 z ζ 0 ζ. = X \ {{z 1 = 0} {ζ 1 = 0} {[0:1], [0:1])}} where π :X ss 0 Xss 0 /T = Y = CP 1 s gve by the map π :[z 0 :z 1 ], [ζ 0 :ζ 1 ]) [z 0 ζ 1 :z 1 ζ 0 ]. Furthermore, a calculato shows that X = cl Γπ = {z 0 ζ 1 1 z 1 ζ 0 0 = 0} X CP 1 f [ 0, 1 ] are the homogeeous coordates of CP 1. A further aalyss of the geometry of X ad Y reveals that Y 0 = Y. Note that we have p X π 1 [ζ 0, ζ 1 ]) ) = cl π 1 [ζ 0, ζ 1 ]) ) for all [ζ 0, ζ 1 ] such that ζ 0,1 0. For [ζ 0 ] = [1:0] ad ζ 1 = [0:1], we have a proper cluso for all {0, 1}. p X π 1 [ζ ]) ) cl π 1 [ζ ]) ) IV. The Fber Probablty Measure Sequece Tame Case) Let U = {U j } j be a fte cover of Y the, after havg chose a fte refemet U of U for coveece set U = U ), t follows by Secto II that there exsts a tame collecto { )} s so that U j π X ) = Y. By chagg the dex set J of the fte cover U we ca always assume that U j = U. 19

22 IV FIBER PROBABILITY MEASURE SEQUENCE Let Y 0 be as Secto IV.1 ad set s :X Y 0 R 0, s x) := ππ 1 x)) s d [π y ] where π 1 πx)) d [π y] deotes the fber tegral of ω k wth respect to the k-fberg π :X 0 Y 0 as defed the work of J. Kg cf. [K]). Remark IV.. Sce we have s ω π 1 y) ) k π 1 y) π 1 y) t follows by the compactess of π 1 y) where y Y, that s < for all y Y 0 Y. ζ p X ) s ω π 1 y) ) k The ext step s to defe a sequece of collectos of fber probablty destes attached to U = {U } as follows. Defto IV.1. Let U = {U } ad { s )} be as above,.e. U π X ), the defe a sequece of collectos of fber dstrbuto destes o π 1 U ) X 0 by φ :x φ x) = s x) s x). I the sequel, the termology { } { } φ U = φ wll be used. Furthermore, { } φ U wll be referred to as the collecto of fber dstrbuto destes assocated to { s } Havg defed { φ U }, t s self-evdet to troduce Defto IV.. Let { } φ U be a sequece of collectos of fber dstrbuto destes assocated to a tame collecto { s }, the defe a sequece of collectos of fber probablty measures over Y 0 by ν y):a ν y) A) := A dν := A. φ d [π y ] where y U Y 0 ad A π 1 y) measurable. As Defto IV.1, set { } { } ν U = ν ad refer to { } ν U as the collecto of fber probablty measures assocated to { s }. We complete our deftos wth Defto IV.3. Let { } φ U be a sequece of collectos of fber dstrbuto destes assocated to a tame collecto { s }, the defe a sequece of collectos of cumulatve fber probablty destes over Y 0 by D y, ):t D y, t) := 0 {φ t} π 1 πx)) d [π y ]

23 IV FIBER PROBABILITY MEASURE SEQUENCE where y U Y 0. As the aforemetoed deftos, we set { } { } D U = D ad refer to { } D U as the collectos of cumulatve fber probablty destes assocated to { s } I Secto VI we wll gve a prove of the followg two covergece results. Theorem 3. [Uform Covergece of the Tame Measure Sequece] For for every tame collecto { s } there exsts a fte cover U of Y wth U π X ) so that the collecto of fber probablty measures { } { } ν U assocated to s coverges uformly o Y 0 to the fber Drac measure of µ 1 ) X 0,.e. for every I ad every f C 0 X), we have y ) π 1 y) f dν y) f red uformly o U Y 0. Theorem 4. [Uform Covergece of the Tame Dstrbuto Sequece] For every t R ad every tame collecto { s } there exsts a fte cover U of Y wth U π X ) so that the collecto of cumulatve fber probablty destes { D U, t) } assocated to { s } coverges uformly o Y 0 to the zero fucto o Y 0,.e. for every I we have y D y, t) ) 0 uformly o U Y 0 π X ).. 1

24 IV FIBER PROBABILITY MEASURE SEQUENCE IV.3 π :X ss The Fber Probablty Measure Sequece No-Tame Case) Let s ) be a sequece of T-egesectos so that O 1). Moreover, let Y = Xss /T be the projecto map attached to the Hlbert quotet assocated to the level subset µ 1 ) ad π :X ss 1 Y = X ss 1 /T the correspodg Hlbert quotet assocated to the level subset µ 1 1 ). The am of ths subsecto s to defe a sequece ν ) of fber probablty measures over the base Y 0 by ν y):a ν y) A) := A s s d [π y]. However, sce X s ) = {x X:?s x) 0} moves as, the above measure s ot well defed o all of Y: For example f y Y s so that π 1 y) {s = 0}, the the above measure s ot defed for at y. I some cases, t s however possble to crcumvet ths problem. For ths we wll frst troduce the followg defto. Defto IV.4. [Removable Sgularty] A pot y Y s defed to be a removable sgularty of order N 0 for the fber measure ν f there exsts a ope eghborhood U y Y of y, a sequece f y, ) of local holomorphc fuctos f O U y ) ad N 0 N so that ŝ fy, := s π f 1 y, defes a local holomorphc secto o π 1 U y ) for all N 0 whch does ot vash detcally o π 1 y ) for all y U y. I the sequel, we wll deote the set of all removable sgulartes of order N 0 by R N0. Remark IV.3. The local exteso ŝ fy, s aga a -egesecto. As a corollary of the defto, we deduce Corollary IV.1. If y Y 0 s a removable sgularty for ν of order N 0 N, the after havg shruke U y approprately, the quotet ŝ fy, ŝ fy, s depedet of the local scalg fuctos f y, ),.e. we have over U y Y 0 R N0. ŝ f 0 y, y, ŝ f 0 y, = ŝ f 1 ŝ f 1 y, Proof. Frst of all, sce y s cotaed the ope set Y 0 R N0, we ca assume that U y Y 0. Let fy, ), f y, O U y ) where {0, 1}. As Defto IV.4, we have ŝ f y, = s π fy,, 1. Throughout ths proof, we use the abbrevato fy, = f, for {0, 1}. Usg ths otato, t follows that ŝ f0, = π f 1, f0, 1 ) ŝ f 1, IV.1)

25 IV FIBER PROBABILITY MEASURE SEQUENCE where h 0,1, := f 1, f 1 0, yelds a sequece of meromorphc fucto defed o U y. We clam that after havg shruke U y, each h 0,1, s bouded from above ad below away form zero o U y. For ths ote the followg: Sce ŝ f1, π 1 y) 0 there exsts oe x π 1 y) so that ŝ f1, x) 0 ad hece ŝ f1, x) > 0. I partcular, we fd a ope eghborhood V X 0 of x so that ŝ f1, x ) c > 0 IV.) for all x V. By cotuty we also have ŝ f0, x ) C < IV.3) for all x V. Sce V X 0 ad sce π X 0 :X 0 Y 0 s a k-fberg ad hece a ope map cf. [Loj], p. 97 f.), we ca assume after havg shruke V ad U y approprately) that π V) = U y. Combg IV., IV.3 ad IV.1 we deduce h 0,1, y ) C < for all y U y. Reversg the roles of f 0 ad f 1 shows after havg shruke U y aga) that h 0,1, y ) c > 0. Hece, the meromorphc fucto h 0,1, = f 1, f0, 1 s bouded from above ad below away from zero o U y. As the base Y s assumed to be ormal, we ca apply Rema s Exteso Theorem cf. [Gr-Re], p. 144) order to deduce that h 0,1, yelds a o-vashg holomorphc fucto o U y. All all we, deduce ŝ f 0 y, ŝ f 0 y, π h 0,1, π h 0,1, = ŝ f 1 y, ŝ f 1 y, over U y as clamed. Q. E. D. I ths sese, the quotet s s ca be uquely exteded oto R N0 Y 0. Assume ow that Y 0 R N0. Usg the depedece of ŝ fy, ŝ fy, of the chose sequece f y, ), f y, O U y ) show Corollary IV.1, we ca defe Defto IV.5. Let s ) be sequece of -egesectos whose rescaled weghts approxmate, the defe a sequece of fber dstrbuto destes over Y 0 R N0 by φ :x φ x) := ŝ fy, x) ŝ fy, x) where f y, ) s a sequece of holomorphc fuctos f y, O U y ) as Defto IV.4. As the tame case we troduce the followg two deftos. Defto IV.6. Let φ ) be the sequece of fber dstrbuto destes assocated to s ) as defed Defto IV.5, the we defe a sequece of fber probablty measures parametrzed over Y 0 R N0 by ν y):a ν y) A) := 3 A dν := A φ d [π y ]

26 IV FIBER PROBABILITY MEASURE SEQUENCE for A π 1 y) measurable where y Y 0 R N0. Defto IV.7. Let s ) be sequece of -egesectos whose rescaled weghts approxmate, the we defe a sequece of cumulatve fber probablty destes over Y 0 R N0 by D y, ):t D y, t) := d [π y ] where y Y 0 R N0. {φ t} π 1 πx)) The am of Secto VII s to gve a proof of the followg two covergece results. Theorem 5.b. [Uform Covergece of the Ital Dstrbuto Sequece] For fxed t R the sequece D, t)) coverges uformly o Y 0 R N0 to the zero fucto. Theorem 6.b. [Uform Covergece of the Ital Measure Sequece] Let f C 0 X) the the sequece y π 1 y) f dν y) coverges uformly over Y 0 R N0 to the reduced fucto f red. We close ths secto by showg that geeral there exsts o N 0 N so that Y = R N0. Furthermore, the example shows that eve the extreme case R N0 = for all N 0 N s possble. Example IV.. Let X CP k CP k equpped wth the T = C acto gve by t. [ζ 0 :...:ζ k ], [z 0 :...:z k ]) = [ζ 0 :...:ζ k ], [t 1 z 0 :...:t 1 z k 1 :z k ] ). I the sequel, we wll cosder the T-learzato L = p 0 O CP 1) k p 1 O CPk 1) of the T- acto o X duced by the trval T-acto o the frst factor O 1) CP k CP k gve by ) t.[ζ 0 :...:ζ k :ζ] = [ζ 0 :...:ζ k :ζ] where we have used the detfcato O 1) CP k the secod factor O CP k 1) CP k gve by = CP k+1 \ {[0:...:0:1]} ad the T-acto o t.[z 0 :...:z k :ζ] = [t 1 z 0 :...:t 1 z k 1 :z k :ζ]. A calculato shows that the = 0-level of the assocated momet map s gve by the set µ 1 0) = CP k {[0:...:0:1]} 4

27 IV FIBER PROBABILITY MEASURE SEQUENCE ad t s also drect to verfy that oe ca detfy Y = X ss =0 /T = CP k where X ss 0 = CP k C k. Furthermore, each fber of the quotet map π s somorphc to C k equpped wth the verse dagoal acto. We wll ow cosder the sequece s ) whose rescaled weghts coverge to = 0 defed by Note that we have k 1 s = =0 ζ z z 1 k H 0 X, L ). s π 1 [0:...:0:1]) 0 for all N. Usg the homogeous stadard coordates ζ = ζ ζ k, z = z z k, 0 k 1 o the ope subset U k,k = {[z], [ζ]): ζ k 0, z k 0} X ss 0 CP k CP k, the restrcto π U k,k V k = C k of the quotet map π :X ss 0 CPk s gve by the projecto map π U k,k :U k,k = C k C k C k. Wth respect to ths trvalzato the sequece s U k,k s gve by k 1 s U k,k = ζ, z where π 1 [0:...:0:1]) = {0} C k. To shorte otato, we wll just wrte s U k,k = s throughout the rest of ths example ad set ζ = ζ, z = z for all 0 k 1. Assume ow that ζ 0 = [0:...:0:1] s a removable sgularty for the fber measure ν we wll fx heceforth),.e. there exsts a o-vashg holomorphc fucto f O U) defed o a ope eghborhood U C k of ζ 0 so that ŝ := s π f 1 s holomorphc ad does ot vash detcally o π 1 ζ ) for all y U. Note that, after havg shruke U, we ca assume that ζ 0 s a solated zero of the fucto f. I partcular, the restrcto ŝ π 1 ζ ) defes a o-vashg lear oe form o π 1 ζ ) = C k for all ζ U. Usg ths, t follows that for each sequece ζ m ) m U covergg to ζ 0, the sequece of oecodmesoal subspaces C k = π 1 ζ m ) gve by H ζ m ) = { x C k : ŝ ζ m ) = 0 } must coverge to a uquely defed oe-codmesoal subspace whch s depedet of the choce of ζ m ) m. However, ths s a cotradcto to the equato ŝ = s π f 1 ad the fact that f s o-zero o U \ {ζ 0 }. For example, cosder the collecto of sequeces gve by { ζ m } =0 = { 0,..., m 1,..., 0 ) m }, the we have H ζm ) = {x C k ) } { } : ŝ ζ m = 0 x C k : z = 0 so the lmt s ot depedet of the chose sequece. Hece, we deduce a cotradcto ad t follows that ζ 0 s a ot a removable sgularty for ay N 0 N the sese of Defto IV.4,.e we have ζ 0 / R N0 for all N 0. 5

28 IV FIBER PROBABILITY MEASURE SEQUENCE Furthermore, by slghtly chagg the above sequece s ), we ca show that R N0 = for each N 0 : Choose a dese sequece ζ ) the quotet Y = CP k ad let Φ ), Φ Aut Y) be a sequece of projectve trasformatos so that Φ ζ 0 ) = ζ. Defe a ew sequece of egesectos by ad cosder the sequece gve by s [ζ], [z]) := s 1 [Φ ζ)], [z]) s := s. It s drect to see that R N0 = because for each ope eghborhood U of ay pot y Y, the subset U {ζ } s dese by costructo ad ζ s o-removable. =1 As a cosequece of ths example we coclude Remark IV.4. There are examples of approxmatg sequeces s ) so that R N0 all N 0 N 0. = for As a further-reachg questo, oe could ask whether the set of all -approxmatg sequeces of -egesectos s ), wth the property that R N0 = for all N 0, s th as a subset of =0 H0 X, L ). It turs out that there s o deftve aswer to ths questo: I the cotext of Example II.1, oe ca show that each sgularty of s H 0 X, L ) s removable ad hece Y = R N0 for all N 0 N ad ay choce of s ). O the other had, f k = Example IV., t turs out that a -egesecto s H 0 X, L ), whch has bee radomly chose wth respect to a choce of a Lebesgue measure o H 0 X, L ) duced by a choce of a bass), has almost surely at least oe o-removable sgularty y Y. Moreover, f oe radomly chooses a -approxmatg sequece of -egesectos ths settg, t turs out that the set {y } N s almost surely dese Y. 6

29 V THE K-FIBERING Π: X Ỹ V The k-fberg Π: X Ỹ V.1 Costructo of the k-fberg Π: X Ỹ Let π : X Y be the holomorphc map betwee the purely dmesoal varetes X ad Y as defed Secto IV.1 ad recall that there exts a Zarsk-dese subset Y 0 Y so that all fbers π 1 y) for y Y 0 are purely k-dmesoal, compact subvaretes ot ecessarly rreducble. Recall that we have assumed X to be ormal ad hece, t follows cf. [He-Hu1], p. 14) that Y s ormal ad therefore, partcular, the ope subset Y 0 as well. By [Bar1] the followg s kow the above cotext: There exsts a holomorphc map ϕ π : Y 0 C k X ) to the cycle space C k X ) of all compact k-dmesoal cycles C = I C, N, C X globally rreducble subspaces of X of dmeso k so that the support ϕ π y) of the cycle C y := ϕ π y) for y Y 0 s equal to the set theoretc fber π 1 y),.e. we have C y = π 1 y) for all y Y 0. V.1) Furthermore, sce X ad Y are compact, there exsts cf. [Bar1]) a proper modfcato σ : Ỹ Y wth ceter Y \ Y 0, a proper modfcato Σ: X X wth ceter π 1 Y \ Y 0 ) ad a surjectve holomorphc map Π: X Ỹ so that the followg dagram commutes: X Y π Σ σ X Ỹ Π The compact, complex space Ỹ := cl Ỹ s gve by { } y, C) Y 0 C k X ): ϕ π y) = C Y C k X ) V.) ad the holomorphc map σ : Ỹ Y s defed by σ := p Y Ỹ. Moreover, f X Ck X ) X deotes the uversal space defed by X := { C, x ) C k X ) X : x C }, the X s compact ad gve by X = Y X) Ỹ X ) where Σ s the restrcto of the projecto p :Y C k X ) X X to X. Note that the above costructo, whose detals ca be foud [Bar1], mples that the fber Π 1 y, C) X detfes wth C X. Usg ths detfcato, we wll smply wrte Π 1 y, C) = C The support C of a cycle C = I C s defed by C = I C. 7

30 V THE K-FIBERING Π: X Ỹ heceforth. Remark V.1. Note that Π: X Ỹ s a holomorphc map whose fbers are purely k- dmesoal by costructo. Furthermore, X ad Ỹ are both purely dmesoal where dm C X = dmc X ad dm C Ỹ = dm C Y whch s a well kow fact of the theory of proper modfcatos, cf. [Gr-Re], p. 14). Moreover, we have the followg lemma. Lemma V.1. The support Π 1 y, C) = C s T-varat subset of π 1 y). Proof. By the commutatvty of the prevous dagram, we deduce that C π 1 y), hece t remas to verfy that C s T-varat. I order to prove ths, we ca proceed as follows: Sce the set { y, C) Y 0 C k X ): ϕ π y) = C } s Eucldea dese Ỹ, we ca choose a sequece y ) Y 0 so that y, C y ) y, C) where C y are the cycles whose uderlyg sets are equal to π 1 y ) by property V.1. The T-varace of the lmt cycle C follows by the followg reasog: Let x = t.x 0 T.C where x 0 C. The, sce C C meas covergece the Hausdorff topology of the uderlyg support, there exsts a sequece x C so that x x 0. Sce C y = π 1 y) for all y Y 0, t follows that C y = π 1 y ) for all N. Usg the T-varace of π 1 y) cf. Remark IV.1), we deduce that t.x C y. By the cotuty of the acto t follows that t.x t.x 0. So t.x ) s a coverget sequece wth lmt t.x 0 where t.x C ad C C. By the defto of the Hausdorff topology, t the follows that t.x 0 C ad hece T.C C whch proves the clam T.C = C. Q. E. D. We close ths secto wth the followg remark ad example. Remark V.. I geeral cf. Example V.1) C s a proper subset of π 1 y). Example V.1. Let X = CP 3 equpped wth the T = C acto gve by ad cosder the Hlbert quotet t.[z 0 :z 1 :z :z 3 ] = [t 1 z 0 :t z 1 :t z :z 3 ] π :X ss 0 = CP 3 \ {[1:0:0:0]} {z 0 = z 3 = 0}) Y = CP assocated to the 0 = -level set of the momet map µ:[z] z z 0 + z 1 + z ). The correspodg projecto map π s gve by π :[z] [ζ] = [z 0 z 1 :z 0 z :z 3 ] where [ζ] Y = CP. Note that all fbers π 1 [ζ]) over Y 0 = CP \ {[0:0:1]} are of pure dmeso 8

31 V THE K-FIBERING Π: X Ỹ oe ad of degree two. Moreover, t s drect to verfy that these fbers ca be parameterzed by γ [ζ] :t [ζ t 1 :ζ 0 t 0 :ζ 1 t 0 :ζ t 0 t 1 ] for ζ Y 0 ad that they are gve as the zero set of the followg system of equatos: ζ z 0 z 1 ζ 0 z 3 = 0, ζ z 0 z ζ 1 z 3 = 0 where ζ Y 0. Let U = { [ζ] CP : ζ 0 } ad set c 0 := ζ 1 ζ 0, c 1 := ζ 1 ζ 1 ad cosder U := Y 0 U whch we ca detfy wth C \ {0}. As metoed before, there exsts a holomorphc map ϕ π :U C1 X ). It turs out that all fbers of π are compact subvaretes of degree CP 3. Hece, t follows that the mage of ϕ π s cotaed the cycle space compoet whch ca be detfed wth the compact coected Chow Varety C 1, CP 3 ) of all 1-dmesoal cycles CP 3 of degree whch tself s realzed as closed varety the projectve space CP ν3,1, for a rgorous defto cf. [Sha]). Recall that the Chow coordates of a cycle C X CP m of degree d ad dmeso k are gve by the coeffcets of the Chow form F C,CP m,.e. by the coeffcets of a polyomal homogeous k + 1 groups ) 0,..., ) m, {0,..., k} of m + 1 determates of degree d modulo multplcato wth a o-vashg complex umber λ C cf. [Sha]). I the sequel, let C c for c C \ {0} ad let F Cc,CP3 be the correspodg Chow form. A calculato shows that ) F Cc,CP 0) 3 0, 0) 1, 0), 0) 3, 1) 0, 1) 1, 1), 1) 3 = c 0 0) 1 1) 0 + c 1 0) 1) 0 + c 0 0) 0 1) 1 + c 1 0) 0 1) + c0 c 1 0) 1) 0 1 1) +c 0 0) 0 0) 1 1) 3 + c1 0) 0 0) 1) 3 + c0 0) 1) 3 0 1) 1 + c 1 0) 1) 3 0 1) + c 0 c 1 0) 1 0) 1) 0 c0 0) 0 0) 3 1) 1 1) 3 c 1 0) 0 0) 3 1) 1) 3 c 0 0) 1 0) 3 1) 0 1) 3 c 1 0) 0) 3 1) 0 1) 3 c 0 0) 0 0) 1 1) 0 1) 1 c 0 c 1 0) 0 0) 1 1) 0 1) c 0 c 1 0) 0 0) 1) 0 1) 1 c 1 0) 0 0) 1) 0 1) so the map ϕ π :C \ {0} C 1, CP 3 ) s gve by ϕ π :C \ {0} c 1, c ) [c 0 :c 0 : c 0 :c 1 :c 1 : c 1 : c 0c 1 : c 0 c 1 : c 0 c 1 : The closure of the graph c 0 c 1 : c 0 : c 0 :c 0 :c 0 : c 1 : c 1 :c 1 :c 1 :0:...:0]. { } c, C): c C \ {0}, ϕ π c) = C C C 1, CP 3 ) 9