Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/mypapers/rokhlin80ru.pdf Äàòà èçìåíåíèÿ: Sat Jan 14 12:38:23 2006 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:02:56 2016 Êîäèðîâêà: Windows-1251

, --
. . , . . . , m- - ( ) m ( m ). . -, . - m- , m- . , , . -- -. .

1. [6] . A ( ) L Cm , U (A) = Cm \ LA L. U (A) U (A) = U (A ) × (C )k , A Cm-k , . - Cm m v1 , . . . , vm : A K (A) . , |A| LA L A;
1991 Mathematics Subject Classification. 55N91, 05B35 (Primary) 13D03 (Secondary). , 99-01-00090, INTAS, 96-0770.
1


2

. . , . .

vI = {vi1 , . . . , vik } (k - 1)- K (A) , (m - k )- LI Cm , zi1 = . . . = zik = 0, |A|. A K (A); U (K ) U A(K ) ( 2). , , ..; . . ( ) Cm . [1] [4] , U (A) , A 21 i dFA , FA , F A, . [18] , H U (A) . - [15, III], H i U (A) ( ) . , ( ) , A. , [15]. [12]. , U (A) . [23]. ~ K , - ~ ~ K . K v1 , . . . , vm , K , "" : ~ vI = {vi1 , . . . , vik } K {v1 , . . . , vm } \ vI K . [13] ~ K - . ~ K K ,




3

. , U (K ) -- ZK , K . ZK - (D2 )m Cm , T m (D2 )m . , ZK i : BT K B T m , B T m m , T BT K , - ( ) k(K ) K . ZK ( U (K )) -. U (K ) Tork[v1 ,...,vm ] k(K ), k k(K ). k(K ) [u1 , . . . , um ], [u1 , . . . , um ] , ui vi k(K ) = k[v1 , . . . , vm ]/I . [12] [23] , [6]. , K (n - 1)- (, K n- ), -- ZK (m + n)- ( , U (K ) ). [5], [6]. ZK , ; . (. [10], [14]) . , (., , [2], [3], [9]), , M , M U (K ) (C )m-n ( K , ). - ZK U (K ) Rm-n T m-n (C )m-n . [11] (


4

. . , . .

), . M 2n T n , T n Cn ; , n- , . , ; [11] (. [7], [8], [5], [6], [19], [20], ). , , . . , (, BT P ) [11] ; . , . , [3], , , [13]. 2. Cm m- z1 , . . . , zm . I = {i1 , . . . , ik } LI (m - k )- , zi1 = . . . = zik = 0. , L{1,...,m} = {0} L = Cm . 2.1. ( ) A LI . A U (A) = Cm \
LI A

LI Cm .

A |A| LI A LI Cm . I J LI |A|, LJ |A|. A zi = 0, U (A) U (A0 ) × C , A0 {zi = 0} C = C \ {0}. ,




5

A U (A) U (A) = U (A ) × (C )k , A Cm-k , . , . A Cm ( ) K (A) m v1 , . . . , vm : , vI = {vi1 , . . . , vik } (k - 1)- K (A), LI |A|. 2.2. 1) A = ; K (A) (m - 1) m-1 . 2) A = {0}; K (A) = m-1 (m - 1)-. , K {v1 , . . . , vm } A(K ) , LI |A| vI = {vi1 , . . . , vik } K . , K K , A(K ) A(K ). , - m Cm . U (K ) = Cm \ |A(K )| A(K ). 2.3. 1) K = m-1 (m - 1)-simplex; U (K ) = Cm . 2) K = m-1 ; U (K ) = Cm \ {0}. 3) K m ; U (K ) Cm , .. zi = zj = 0, i, j = 1, . . . , m. k , . k[v1 , . . . , vm ], vi . 2.4. ( - ) ( k(K )) K k[v1 , . . . , vm ]/I , I = (vi1 ž ž ž vis : {vi1 , . . . , vis } K ) . , , 2, . k(K ) deg vi = 2, i = 1, . . . , m. 2.5. 1) K = m-1 ; k(K ) = k[v1 , . . . , vm ]. 2) K = m-1 (m - 1)-; k(K ) = k[v1 , . . . , vm ]/(v1 ž ž ž vm ).


6

. . , . .

T m Cm ; , A(K ) , U (K ). BT K : (1)
m

BT K = E T

m

×T

m

U (K ),

E T T m - E T m B T m B T m = (CP )m . , BT K BT K B T m U (K ). B T m ( CP ). I = {i1 , . . . , ik } B TIk = B Tik ,...,ik B T m , B T k . 1 2.6. K {v1 , . . . , vm } BT K B T m , B TIk I , vI K . 2.7. K m v1 , . . . , vm . BT K m CP . B T m k[v1 , . . . , vm ] ( k). 2.8. BT K k(K ). i : BT K B T m i : k[v1 , . . . , vm ] k(K ) = k[v1 , . . . , vm ]/I . . K . K v1 , . . . , vm , BT K m CP (. 2.7). H (BT K ) k, 1 k[v1 ] ž ž ž k[vm ]. , H (BT K ) = k [v1 , . . . , vm ]/I , I , 2 , i . , dim K = 0. , K K (k - 1)- vI = {vi1 , . . . , vik }. , K , i H (B T m ) = H (BT K ) = k(K ) = k[v1 , . . . , vm ]/I . 2.6, BT K BT K B Tik ,...,ik B T m . H (BT K 1 B Tik ,...,ik ) = k[v1 , . . . , vm ]/I = k(K vI ), I I 1 vi1 vi2 ž ž ž vik . I
m

m- Rm : = {(y1 , . . . , ym ) Rm : 0 y
i

I

m

1, i = 1, . . . , m}.




7

u

u



u

0
u

u u u u 0 u u u

)

)

. 1. CK . K m v1 , . . . , vm CK , I m . 2.9. (k - 1)- vJ = {vj1 , . . . , vjk } K CJ k I m , m - k yi = 1 , i {j1 , . . . , jk }. /
m

, CK I CJ vJ K .



. CK I m K (. [11, p. 434]). , m-1 (m - 1)- ^ {v1 , . . . , vm }, m-1 m-1 , ^ m-1 vJ m-1 . ^ m-1 I m : vJ : ^ m-1 I m yj = 0 j J yj = 1 j J , / ^ ^ m-1 . m-1 m , . I ^ ^ C C m-1 m-1 I m (1, . . . , 1) ^ ^ C m-1 . C m-1 C m . K I {v1 , . . . , vm }. , K m-1 . CK I m 2.9 ^ C (C K ) K C . 2.10. . 1 ) ) CK , K 3 2- .


8

. . , . .

. , K n- P n , CK P n , [6]. T R
m + m

Cm -

= {(y1 , . . . , ym ) R

m

: yi

0, i = 1, . . . , m}.

Cm Rm (z1 , . . . , zm ) + (|z1 |2 , . . . , |zm |2 ). - (D2 )m = {(z1 , . . . , zm ) C
m

: |zi |

1, i = 1, . . . , m} Cm ,

I m Rm . + UR (K ) Rm U (K )/T m . + , Rm Cm , UR (K ) + " ": UR (K ) = U (K ) Rm . + 2.11. -- ZK Cm , K T m , ZK - - (D2 )m -- CK - - I m , -- T m , CK I m . 2.12. CK UR (K ) Z
K

U (K ).

. 2.11 , . , a = (y1 , . . . , ym ) CK yi1 = . . . = yik = 0, vI = {vi1 , . . . , vik } K , , LI |A(K )|. 2.13. U (K ) -- ZK . . r : UR (K ) CK , U (K ) ZK . . r : UR (K ) CK , (m - 1)- , K . , r. K = m-1 (m - 1)-, UR (K ) = Rm \ {0} + r . 2. , -




9

u

u ? ? & ? ?U T# ! & ? ? & & b Å ? ? && Å ? ? & Å Å Å B ? ? &ÅÅ $$ $ $ X & ? ? ÅÅ $$$ &$$ ?? Å & Å $ e$ E u ? ? ? ?

. 2. r : UR (K ) CK K = m-1 . K (k - 1)- vJ = {vj1 , . . . , vjk } K . , K , r : UR (K ) CK . CJ I m (. 2.9). vJ K , a yj1 = . . . = yjk = 0, yi = 1, i {j1 , . . . , jk }, / U (K ). . 2 CJ a. rJ . r = rJ r . , r . 2.14. 1) K = m-1 (m - 1)- ; ZK (2m - 1)- S 2m-1 . 2) K , n- P n , ZK (m + n)- . , ZP , [6]. 2.15. E T BT K .
m

×T

m

ZK

. r : U (K ) ZK , 2.13, T m U (K ) ZK . BT K = E T m ×T m U (K ). ( ) E T m ×T m ZK BT K = E T m ×T m U (K ). 2.16. i : BT K B T m (. 2.6) p : BT K B T m (. (1)) . , BT K BT K . . : ZK CK - ZK (. 2.11). I = {i1 , . . . , ik } {1, . . . , m} BI -


10

. . , . .

(D2 )m : BI = B1 × ž ž ž × Bm D2 × ž ž ž × D2 = (D2 )m , Bi D2 i I Bi S 1 D2 i I . , / BI (D2 )k × T m-k , k = |I |. , CI = CK (. 2.9), -1 (CI ) = BI . I J , BI BJ . ZK BI , vI K . ( , ZK , K , . [6, 2.4].) vI K BI ZK T m ZK . , BT K = E T m ×T m ZK E T m ×T m BI ( BT P [11, p. 435]). E T m ×T m BI = E T k ×T k (D2 )k × E T m-n , B TIk . , p : BT K B T m E T m ×T m BI B TIk B T m . vI K p : BT K B T m i : BT K B T m . 2.17. U (K ) i : BT K B T m .
2.18. HT m U (K ) T m - U (K ) k(K ). . HT m U (K ) = H E T m ×T m U (K ) H (BT K ). 2.8 2.16.

=

3. U (K ) k[v1 , . . . , vm ]- k(K ) k[v1 , . . . , vm ]: (2) 0R
-h

-- R --

d-

h

-h+1

-- žžž R --

d

-h+1

d-1 -1 -

R0 - k(K ) 0

d0

(, , h m). k[v1 ,...,vm ] k (2), : 0 - R
-h

k[v

1

,...,vm ]

k - ž ž ž - R0 k[v

1

,...,vm ]

k - 0,

Tor-[iv1 ,...,vm ] k(K ), k . k R-i (2) k[v1 , . . . , vm ]--i,j , Tor-[iv1 ,...,vm ] k(K ), k = j Tork[v1 ,...,vm ] k(K ), k k k-, (3) Tork[v1
,...,vm ]

k(K ), k =
i,j

Tor-[i,j k v1

,...,vm ]

k(K ), k




11

k-. , , ( deg vi =2). k- (3) -i + j .
-i,2j 2j k(K ) = dimk Tor-[i,1 ,...,v kv
m

-i

k(K ) = dimk Tor-[iv1 k

,...,vm ]

k(K ), k

]

k(K ), k

; (., , [22]). , -i,2j k(K ) K . 3.1 ( [16], [22]). j Tor-[iv1 ,...,vm ] k(K ), k k
j -i,2j -i,2j

k(K ) t

2j

-

k(K ) t

2j

=
I {v1 ,...,vm }

~ dimk H|I

|-i-1

(KI ) t2|I | ,

KI I.

K ,

, -i,2j k(K ) , K . , Tork[v1 ,...,vm ] k(K ), k , H U (K ) : 3.2. : H U (K ) Tork[v1 ,...,vm ] k(K ), k = . U (K ) - - E T --
i m

(4)

BT K - - B T m , -- . 2.17 , U (K ) U (K ). (4) , C (BT K ) (E T m ) C (B T m ). C 2.8, C (BT K ) = k(K ), i : C (B T m ) = k[v1 , . . . , vm ] k(K ) = C (BT K ) . E T m , C (E T m ) k. , (5) TorC


(B T

m

)

C (BT K ), C (E T m ) Tork[ =

v1 ,...,vm ]

k(K ), k .


12

. . , . .

- (. [21, Th. 1.2]) (4) E2 E2 = TorH TorC TorH


(B T

m

)

H (BT K ), H (E T m )

(B T

m

)

(C (BT K ), C (E T m )).
,...,vm ]

(B T

m

)

(H (BT K ), H (E T m )) = Tork[v1

k(K ), k ,

(5) , E2 , E2 = E . , 3.2 [21] , TorC (B T m ) C (BT K ), C (E T m ) , H U (K ) , . H U (K ) . k(K ) [u1 , . . . , um ] k(K ) = k[v1 , . . . , vm ]/I [u1 , . . . , um ] m , , bideg vi = (0, 2), (6) bideg ui = (-1, 2), d(vi 1) = 0 d(1 ui ) = vi 1,

, d . 3.3. : H U (K ) H k(K ) [u1 , . . . , um ], d , = ( ). . k k[v1 , . . . , vm ]- , 1 1 vi 0. (., , [17, VII,  2]) k[v1 , . . . , vm ]- k: k[v1 , . . . , vm ] [u1 , . . . , um ], d , d (6). Tork[v1 ,...,vm ] ( , ) , Tor k(K ), k = H k(K ) [u1 , . . . , un ], d = [u1 , . . . , um ], d , = k[v1 , . . . , vm ]. 3.2 H U (K ) Tor k(K ), k , . = , U (K ), . 3.4. - U (K ) BT K T m (. (4)) E3 .




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. H U (K ) = H U (K ) E2 = H (BT K ) H (T m ) = k(K ) [u1 , . . . , um ]. , E2 (6). , E3 = H [E2 , d] = H k(K ) [u1 , . . . , um ] = H U (K ) , 3.3. 3.5. ,
vi11 . . . vipp uj1 . . . ujq k(K ) [u1 , . . . , um ],

i1 < . . . < ip , j1 < . . . < iq , H U (K ) . 1 = . . . = p = 1, {vi1 , . . . , vip } K {i1 , . . . , ip } {j1 , . . . , jq } = . . . [6, 5.3]. (. 2.14), K (, , K ), , U (K ) ZK . [6, 2.10], U (K ) U (K )/Rm-n ZK = Rm-n U (K ). A(K ) U (K ) (., , [2], [3], [9]). , n- M , () Zn m U (K )/G. G (C )m , (C )m-n , K , (i- K (i + 1)- ). M 2n. . GR T m-n = m Rm-n G, ÷ : C GR Cm . a Rm-n ÷ ÷-1 (a)/GR - U (K )/G = M ( . [9]). , ÷-1 (a) ZK K = K . 3.6. G C (C )n+1 , = K n-. U (K ) = Cn+1 \ {0}, M = Cn+1 \ {0}/C CP ÷ : Cm R (z1 , . . . , zm ) Cm n.


14 1 2

. . , . . 2n+1

(|z1 |2 + . . . + |zm |2 ), a = 0 ÷-1 (a) S = 2.14).

Z =

K

(.

, K ( , U (K ) ZK ), H U (K ) . 3.7. K n - 1, U (K ) ZK . 1) H U (K ) , 3.3. , H -i,2j (U (K ) , D H -(m-n)+i,2(m-j ) . (n - 1)- K 2) {vi1 , . . . , vin } j1 < . . . < jm-n , {i1 , . . . , in , j1 , . . . , jm-n } = {1, . . . , m}. vi1 ž ž ž vin uj1 ž ž ž uj
m-n

H

m+n

U (K ) H =

m+n

(ZK )

ZK 3) {vi1 , . . . , vin } {vi1 , . . . , vin-1 , vj1 } K , {vi1 , (n - 2), j1 , . . . , jm-n , 2). vi1 ž ž ž vin uj1 ž ž ž uj H
m+n
m-n

+1. (n - 1). . . , vin-1 }
2

= vi1 ž ž ž v

in-1 vj1 uin uj

žžžu

jm-n

U (K ) .

. 1) 2) . [6, 5.1]. 3) , d(vi1 ž ž ž v
in-1 ui
n

uj1 uj2 ž ž ž ujm-n ) = vi1 ž ž ž vin uj1 ž ž ž ujm-n - vi1 ž ž ž vi

n-1

vj1 uin uj2 ž ž ž uj

m-n

k(K ) [u1 , . . . , um ] (. (6)). K - , k(K ) - , k(K ) k[t1 , . . . , tn ] ( n k(K )). , k(K ) -, {1 , . . . , n }, n , i+1 k(K )/(1 , . . . , i ) i = 0, . . . , n - 1. K -, k , k(K ) (, deg vi = 2 k(K )), i = i1 v1 + i2 v2 + . . . + im vm , i = 1, . . . , n. 3.8. , K - J = (1 , . . . , n ) k(K ),




15

. . H U (K ) H k(K )/J [u1 , . . . , um-n ], d , = : bideg vi = (0, 2), bideg ui = (-1, 2); d(vi 1) = 0, d(1 ui ) = i 1,

, K -, U (K ) k(K )/J [u1 , . . . , um-n ] k(K ) [u1 , . . . , um ] 3.3. 3.9. K (m - 1) . k(K ) = k[v1 , . . . , vm ]/(v1 ž ž ž vm ). , H k(K ) [u1 , . . . , um ], d (. 3.3) 1 v1 v2 ž ž ž vm-1 um , . deg(v1 v2 ž ž ž vm-1 um ) = 2m - 1, 3.7 , v1 v2 ž ž ž vm-1 um ZK S 2m-1 (. = 2.14 1) ). 3.10. K m . U (K ) Cm ( zi = zj = 0, i, j = 1, . . . , m, . 2.3), k(K ) = k[v1 , . . . , vm ]/I , I vi vj , i = j . 3.3 3.5, H U (K ) - vi1 ui2 ui3 ž ž ž uik k(K ) [u1 , . . . , um ] , k 2, ip = iq p = q . -1 k m m-1 , m k k ( ui1 ž ž ž uik ). deg(vi1 ui2 ui3 ž ž ž uik ) = k + 1, dim H 0 U (K ) = 1, dim H dim H
k+1 k+1

H 1 U (K ) = H 2 U (K ) = 0,
m-1 k -1

U (K ) = m U (K ) = 0,

-

m k

,

2

k

m,

k > m.

. , m = 3 6 vi uj , i = j , 3 vi uj = vj ui , 3 v1 u2 u3 , v2 u1 u3 , v3 u1 u2 , v1 u2 u3 - v2 u1 u3 + v3 u1 u2 = 0. , dim H 3 U (K ) = 3, dim H 4 U (K ) = 2, .


16

. . , . .

3.11. K m- (m 4). , , - ZK m + 2, U (K ) ZK . k(K ) = k[v1 , . . . , vm ]/I , I vi vj , i = j + 1 ( vm+i = vi vi-m = vi ). [6]. 1 k = 0 m + 2; k 0 k = 1, 2, m m + 1; dim H U (K ) = (m - 2) m-2 - m-2 - m-2 3 k m - 1.
k-2 k -1 k-3

, m = 5 5 H 3 U (K ) , vi ui+2 k(K ) [u1 , . . . , u5 ], i = 1, . . . , 5, 5 H 4 U (K ) , vj uj +2 uj +3 , j = 1, . . . , 5. 3.7, vi ui+2 vj uj +2 uj +3 H 7 U (K ) ( , ) , {i, i + 2, j, j + 2, j + 3} = {1, 2, 3, 4, 5}. , [vi ui+2 ] ( ) [vj uj +2 uj +3 ] , [vi ui+2 ] ž [vj uj +2 uj +3 ] .
[1] . . , , . 5 (1969), 227-231. [2] M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics 93, BirkhÅ auser, Boston Basel Berlin, 1991. [3] V. V. Batyrev, Quantum Cohomology Rings of Toric Manifolds, Journ? ees de G? ? eometrie Alg? ebrique d'Orsay (Juillet 1992), Ast? erisque 218, Soci? ete Math? ematique de France, Paris, 1993, pp. 9-34; http://xxx.lanl.gov/abs/alg-geom/9310004. [4] E. Brieskorn, Sur le groupes de tresses, in: S? eminare Bourbaki 1971/72, Lecture Notes in Math. 317, Springer-Verlag, Berlin-New York, 1973, pp. 21-44. [5] . . , . . , , , . 53 (1998), . 3, 195-196. [6] . . , . . , , . 225 (1999), 96-131; http://xxx.lanl.gov/abs/math.AT/9909166. [7] . . , . , , . 53 (1998), . 2, 139-140. [8] V. M. Buchstaber and N. Ray, Tangential structures on toric manifolds, and connected sums of polytopes, preprint UMIST, Manchester, 1999. [9] D. A. Cox, Recent developments in toric geometry, in: Algebraic geometry (Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9-29, 1995), J. Kollar, (ed.) et al. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62 (pt.2), 389-436 (1997); available at http://xxx.lanl.gov/abs/alg-geom/9606016. [10] . Danilov, , . 33 (1978), . 2, 85-134. [11] M. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. Journal 62, (1991), no. 2, 417-451.




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[12] C. De Concini and C. Procesi, Wonderful models of subspace arrangements, Selecta Mathematica, New Series 1 (1995), 459-494. [13] M. De Longueville, The ring structure on the cohomology of coordinate subspace arrangements, preprint, 1999; http://www.math.tuberlin.de/~ziegler. [14] W. Fulton, Introduction to Toric Varieties, Princeton Univ. Press, 1993. [15] . , . , , , , 1991. [16] M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in: Ring Theory II (Proc. Second Oklahoma Conference), B. R. McDonald and R. Morris, editors, Dekker, New York, 1977, pp. 171-223. [17] . , , , , 1966. [18] P. Orlik and L. Solomon, Combinatorics and Topology of Complements of Hyperplanes, Invent. Math. 56 (1980), 167-189. [19] . . , y - , . 54 (1999), . 5, 169-170. [20] T. E. Panov. Hirzebruch genera of manifolds with torus action, preprint, 1999; http://xxx.lanl.gov/abs/math.AT/9910083. [21] L. Smith, Homological Algebra and the Eilenberg-Moore Spectral Sequence, Transactions of American Math. Soc. 129 (1967), 58-93. [22] R. Stanley, Combinatorics and Commutative Algebra, Progress in Math. 41, BirkhÅ auser, Boston, 1983. [23] S. Yuzvinsky, Smal l rational model of subspace complement, preprint, 1999; http://xxx.lanl.gov/abs/math.CO/9806143. - , 119899, E-mail address : tpanov@mech.math.msu.su buchstab@mech.math.msu.su