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1052:
753:
252:
884:
324:
646:
122:
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169:
1226:
M. Brion, C. Procesi, Action d'un tore dans une variété projective, in
Operator algebras, unitary representations, and invariant theory (Paris 1989), Prog. in Math. 92 (1990), 509–539.
511:
928:
1097:
785:
353:
936:
816:
276:
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467:
1223:
A. Bialynicki-Birula, "Some
Theorems on Actions of Algebraic Groups," Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480–497
657:
1280:
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366:
does not have finite dimension, the existence of such a decomposition is tricky but one easy case when decomposition is possible is when
180:
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285:
586:
84:
1304:
520:
133:
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The decomposition exists because the linear action determines (and is determined by) a linear representation
1319:
476:
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1149:
1120:
893:
1314:
1174:
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70:
of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus
514:
1060:
43:. In differential geometry, one considers an action of a real or complex torus on a manifold (or an
757:
395:
67:
25:
54:
algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a
1192:
1187:
Altmann, Klaus; Ilten, Nathan Owen; Petersen, Lars; Süß, Hendrik; Vollmert, Robert (2012-08-15).
1047:{\displaystyle S=\bigoplus _{m_{0},\dots m_{n}\geq 0}S_{m_{0}\chi _{0}+\dots +m_{n}\chi _{n}}.}
1210:
21:
1250:
329:
1202:
1103:, then this is the usual decomposition of the polynomial ring into homogeneous components.
794:
261:
373:
29:
411:
1154:
1293:
55:
51:
1127:
1144:
748:{\displaystyle \chi _{i}(t)=t_{1}^{\alpha _{i,1}}\dots t_{r}^{\alpha _{i,r}},}
1111:
The Białynicki-Birula decomposition says that a smooth projective algebraic
1238:
1246:
44:
1175:"Konrad Voelkel » Białynicki-Birula and Motivic Decompositions «"
58:(for example, orbit closures that are normal are toric varieties).
1197:
1206:
32:
on the variety. A variety equipped with an action of a torus
258:-invariant subspace called the weight subspace of weight
1254:
247:{\displaystyle V_{\chi }=\{v\in V|t\cdot v=\chi (t)v\}}
1063:
939:
896:
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797:
760:
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479:
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394:; see below for an example). Alternatively, one uses
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87:
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370:is a union of finite-dimensional representations (
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163:
116:
879:{\displaystyle x_{0}^{m_{0}}\dots x_{r}^{m_{r}}}
319:{\displaystyle \pi :T\to \operatorname {GL} (V)}
74:is acting on a finite-dimensional vector space
641:{\displaystyle t\cdot x_{i}=\chi _{i}(t)x_{i}}
1274:
117:{\displaystyle V=\bigoplus _{\chi }V_{\chi }}
8:
469:be a polynomial ring over an infinite field
241:
197:
78:, then there is a direct sum decomposition:
574:{\displaystyle t=(t_{1},\dots ,t_{r})\in T}
164:{\displaystyle \chi :T\to \mathbb {G} _{m}}
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171:is a group homomorphism, a character of
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506:{\displaystyle T=\mathbb {G} _{m}^{r}}
357:diagonalizable linear transformations
7:
1235:
1233:
1126:It is often described as algebraic
923:{\displaystyle \sum m_{i}\chi _{i}}
1253:. You can help Knowledge (XXG) by
14:
822:-weight vector and so a monomial
359:, upon extending the base field.
1237:
1107:Białynicki-Birula decomposition
1092:{\displaystyle \chi _{i}(t)=t}
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780:{\displaystyle \alpha _{i,j}}
1189:The Geometry of T-Varieties
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1232:
890:-weight vector of weight
16:In algebraic geometry, a
400:Hilbert-space direct sum
62:Linear action of a torus
348:{\displaystyle \pi (T)}
1150:Equivariant cohomology
1121:cellular decomposition
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398:; for example, uses a
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355:consists of commuting
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811:{\displaystyle x_{i}}
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515:algebra automorphisms
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271:{\displaystyle \chi }
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383:{\displaystyle \pi }
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1310:Algebraic varieties
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462:{\displaystyle S=k}
396:functional analysis
1300:Algebraic geometry
1140:Sumihiro's theorem
1115:-variety admits a
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1216:978-3-03719-114-9
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22:algebraic variety
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1305:Algebraic groups
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1247:geometry-related
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30:algebraic torus
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1320:Geometry stubs
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1155:monomial ideal
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1249:article is a
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527:
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516:
513:act on it as
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69:
68:linear action
61:
59:
57:
56:toric variety
53:
48:
46:
42:
40:
35:
31:
27:
23:
19:
1315:Morse theory
1255:expanding it
1244:
1229:
1188:
1169:
1128:Morse theory
1125:
1116:
1112:
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1100:
1056:
887:
819:
790:
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405:
404:
367:
363:
361:
281:
255:
172:
126:
75:
71:
65:
49:
38:
37:
36:is called a
33:
26:group action
18:torus action
17:
15:
1207:10.4171/114
1145:GKM variety
787:= integers.
1294:Categories
1161:References
791:Then each
390:is called
1198:1102.5760
1066:χ
1031:χ
1014:⋯
1002:χ
978:≥
965:…
948:⨁
930:. Hence,
912:χ
898:∑
852:…
763:α
726:α
712:…
695:α
663:χ
611:χ
594:⋅
566:∈
547:…
441:…
378:π
334:π
326:and then
305:
299:→
290:π
266:χ
227:χ
218:⋅
204:∈
190:χ
147:→
138:χ
110:χ
100:χ
96:⨁
1134:See also
1119:-stable
1099:for all
1057:Note if
517:by: for
392:rational
45:orbifold
41:-variety
406:Example
1213:
651:where
473:. Let
408:: Let
127:where
52:normal
28:of an
20:on an
1245:This
1193:arXiv
886:is a
818:is a
24:is a
1251:stub
1211:ISBN
1203:doi
362:If
47:).
1296::
1209:.
1201:.
1191:.
1130:.
1123:.
402:.
302:GL
254:,
66:A
50:A
1282:e
1275:t
1268:v
1257:.
1219:.
1205::
1195::
1177:.
1117:T
1113:T
1101:i
1087:t
1084:=
1081:)
1078:t
1075:(
1070:i
1042:.
1035:n
1025:n
1021:m
1017:+
1011:+
1006:0
996:0
992:m
987:S
981:0
973:n
969:m
962:,
957:0
953:m
944:=
941:S
916:i
906:i
902:m
888:T
870:r
866:m
860:r
856:x
845:0
841:m
835:0
831:x
820:T
804:i
800:x
773:j
770:,
767:i
743:,
736:r
733:,
730:i
720:r
716:t
705:1
702:,
699:i
689:1
685:t
681:=
678:)
675:t
672:(
667:i
634:i
630:x
626:)
623:t
620:(
615:i
607:=
602:i
598:x
591:t
569:T
563:)
558:r
554:t
550:,
544:,
539:1
535:t
531:(
528:=
525:t
499:r
494:m
489:G
484:=
481:T
471:k
457:]
452:n
448:x
444:,
438:,
433:0
429:x
425:[
422:k
419:=
416:S
368:V
364:V
343:)
340:T
337:(
314:)
311:V
308:(
296:T
293::
278:.
256:T
242:}
239:v
236:)
233:t
230:(
224:=
221:v
215:t
211:|
207:V
201:v
198:{
195:=
186:V
175:.
173:T
157:m
152:G
144:T
141::
106:V
92:=
89:V
76:V
72:T
39:T
34:T
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
