Документ взят из кэша поисковой машины. Адрес оригинального документа : http://higeom.math.msu.su/people/taras/talks/2009khabarovsk-talk.pdf Дата изменения: Sun Apr 8 16:04:10 2012 Дата индексирования: Sun Apr 10 01:12:59 2016 Кодировка:

-
. . . . .

XXXIV - . . 25-30 2009,


- . Rn -- . P = {x Rn : (a i, x ) + bi : ) dim P = n; ) ( , P ); ) P ; ) Hi = {(a i, x ) + bi = 0}, 1 . i m, 0 1 i m}, a i Rn, bi R.

2


P n- m Fi = {x P : (a i, x ) + bi = 0} = P Hi a i, for 1 i m.

P . , . .

3


P P = {x : AP x + b P 0}, AP = (aij ) m в n a i, b P -- - bi. x AP x + b P P Rm = {y Rm : yi 0}. ZP ZP -Z Cm

i

iP : Rn - Rm,

(z1, . . . , zm)


P iZ T m- .
4

- Rm

iP

(|z1|2, . . . , |zm|2)


1. ZP T m- iZ : ZP Cm. . 1) iP (Rn) Rm m - n m cj k (yk - bk ) = 0, 1 j m - n; k=1 2, 2) yk |zk | ZP m - n :
m k=1

cj k |zk |2 - bk

(

)

= 0 1 j m - n.

3) , 2) , .. ZP . ZP - , P .
5


­. y = (y1, . . . , ym) Rm T (y ) = {t = (t1, . . . , tm) T m : ti = 1, yi = 0} T m. Cm Rm в T m/ , (y , t ) = (y , t ) y = y and t -1t T (y ). iZ : ZP Cm ZP P в T m/ in Rm в T m/. 1. ZP P . T m- ZP [Bosio­Meersseman].
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. K -- () [m] = {1, . . . , m}. = {i1, . . . , ik } K -- ; K . 1. P KP = = {i1, . . . , ik } : Fi1 . . . Fik = P
{ }

-- ( ) . |KP | = S n-1.

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- . D2 C -- . {1, . . . , m} B := {(z1, . . . , zm) (D2)m : |zi| = 1 i } / = (D2)| | в (S 1)m-| |. - ZK :=

K

B (D2)m.

2. ZK T m: ZK

2 - (D)m



,

cone K - Im K -- ; = {i1, . . . , ik } e = (1, . . . , m), i = 0 i i = 1 i . /
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K = KP P , cone K P , ZKP ZP ! , 3. a) |K | = S n-1 ( m ). ZK (m + n)-; ) K -- . ZK \ Z , Z = T m. 2. Z n = S 2n+1. n = 1 S 3 = D2 в S 1 S 1 в D2 D2 в D2.

9


. · , P ; · P в T m/ | cone K | в T m/;
· K B (D2)m.

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K = KP , ZK «» K . 1. ZK , K , ? , ZP , ( , S 1). , ­ [Bosio­Meersseman].
11


. Cm L = {(z1, . . . , zm) Cm : zi1 = . . . = zik = 0}, = {i1, . . . , ik }. Cm K m . U (K ) = Cm \

K /

L .

4. T m- . U (K ) - ZK .


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. K -- m . Z[K ] = Z[v1, . . . , vm]
/(

/ vi1 · · · vik : {i1, . . . , ik } K

)

-- ( ­) K , deg vi = 2.

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. 1. [-.] () , H (ZK ; Z) = TorZ[v ,...,v ](Z[K ], Z) = H [u1, . . . , um] Z[K ]; d , dui = vi, dvi = 0 1 H p(ZK ) = i

-i+2j =p

[

1

m

]

m. , TorZ[v ,...,v ](Z[K ], Z). m 1
-i,2j

2. H -i,2j (ZK , Z) =

| | =j

H j -i-1(K ),

K -- ( K {1, . . . , m}).
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P : 3.
-i,2j (Z ) = Tor-i,2j H j -i-1(P ), (Z[P ], Z) = H P Z[v1 ,...,vm ] | | =j P = i Fi -- .

4. [Goresky­MacPherson] Hi U (K ) =
( )
K

H 2m-2||-i-2(linkK ),

K = { : [m]\ K } ­ , . /

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3. K = m . U (K ) = Cm
\

{zi = zj = 0}

1 i
-- 2, H (U (K )) = H
( m

(S k+1)(k-

1)(m k

).

)

k=2

4. P -- m-, .. KP -- m-. U (KP ) = Cm
\

{zi = zj = 0};
mo d m

i-j =0,1

ZP (m + 2)- , H (ZP ) = H (U (KP )) = H
(m-2
-) k+1 в S m-k+1 )#(k-1)(mk 2) . # (S k=2 16


. , ZP , . , P 3- ( ), ZP [].

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[1] Victor M Buchstab er and Taras E Panov. Torus Actions and Their Applications in Top ology and Combinatorics. University Lecture Series, vol. 24, Amer. Math. So c., Providence, R.I., 2002. [2] Victor M. Buchstab er, Taras E. Panov and Nigel Ray. Spaces of p olytop es and cob ordism of quasitoric manifolds. Moscow Math. J. 7 (2007), no. 2; arXiv:math.AT/0609346. [3] Megumi Harada, Yael Karshon, Mikiya Masuda, Taras Panov, eds. Toric Top ology. Contemp. Math., vol. 460, Amer. Math. So c., Providence, R.I., 2008. [4] . . , . . . . , , 2004. [5] Taras Panov. Cohomology of face rings, and torus actions, in "Surveys in Contemp orary Mathematics". London Math. So c. Lecture Note Series, vol. 347, Cambridge, U.K., 2008, pp. 165­ 201; arXiv:math.AT/0506526.
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