3640:
2953:
1451:
4206:
1799:
3479:
2779:
3080:
2643:
1953:
2659:(1). (In fact, the above isomorphism is part of the usual correspondence between Weil divisors and Cartier divisors.) Finally, the dual of the twisting sheaf corresponds to the tautological line bundle (see below).
2539:
1200:
3463:
1292:
3298:
2361:
2036:
1861:
2771:
4092:
252:
3699:
3200:
315:
1682:
2184:
4085:
1595:
1105:
945:
3147:
2428:
1033:
991:
3894:
3393:
2253:
354:
3934:
3335:
3229:
383:
3837:
3754:
1624:
1511:
1483:
123:
3635:{\displaystyle 0\to I\to {\mathcal {O}}_{\mathbb {P} ^{n}}{\overset {x_{i}\mapsto y_{i}}{\longrightarrow }}\operatorname {Sym} {\mathcal {O}}_{\mathbb {P} ^{n}}(1)\to 0,}
4250:
Editorial note: this definition differs from
Hartshorne in that he does not take dual, but is consistent with the standard practice and the other parts of Knowledge.
2130:
1670:
4235:
2100:
2073:
1540:
816:
785:
729:
666:
631:
524:
2948:{\displaystyle \mathbf {Spec} \left({\mathcal {O}}_{\mathbb {P} ^{n}}\right)=\mathbb {A} _{\mathbb {P} ^{n}}^{n+1}=\mathbb {A} ^{n+1}\times _{k}{\mathbb {P} ^{n}}}
758:
698:
2730:
862:
886:
836:
604:
584:
564:
544:
497:
477:
163:
143:
97:
77:
50:
2972:
3756:
this is equivalent to saying that it is a negative line bundle, meaning that minus its Chern class is the de Rham class of the standard Kähler form.
409:
192:
Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as
2566:
1877:
2448:
1446:{\displaystyle {\begin{cases}\phi :\pi ^{-1}(U)\to U\times X\subseteq G_{n}(\mathbb {R} ^{n+k})\times X\\\phi (V,v)=(V,p(v))\end{cases}}}
1130:
3968:(Chern classes of tautological bundles is the algebraically independent generators of the cohomology ring of the infinite Grassmannian.)
3413:
1804:
where on the left the bracket means homotopy class and on the right is the set of isomorphism classes of real vector bundles of rank
606:, varying continuously. All that can stop the definition of the tautological bundle from this indication, is the difficulty that the
3246:
2310:
1970:
1819:
4201:{\displaystyle {\begin{cases}G_{n}(\mathbb {R} ^{n+k})\to \operatorname {End} (\mathbb {R} ^{n+k})\\V\mapsto p_{V}\end{cases}}}
4438:
4380:
4344:
4292:
2735:
202:
181:
since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a
Grassmannian is a
3652:
3156:
4317:
271:
3705:. In practice both the notions (tautological line bundle and the dual of the twisting sheaf) are used interchangeably.
1794:{\displaystyle {\begin{cases}\to \operatorname {Vect} _{n}^{\mathbb {R} }(X)\\f\mapsto f^{*}(\gamma _{n})\end{cases}}}
2135:
4039:
1549:
1059:
899:
1964:: In turn, one can define a tautological bundle as a universal bundle; suppose there is a natural bijection
3937:
3092:
318:
3936:; in other words the hyperplane bundle is the generator of the Picard group having positive degree (as a
2395:
1000:
958:
3863:
3363:
2223:
2132:
It is precisely the tautological bundle and, by restriction, one gets the tautological bundles over all
324:
4313:
3906:
432:
3311:
3205:
359:
4367:
3811:
3702:
2075:
is the direct limit of compact spaces, it is paracompact and so there is a unique vector bundle over
586:, this is already almost the data required for a vector bundle: namely a vector space for each point
266:
4101:
2575:
1886:
1691:
1301:
3844:
3734:
1600:
1491:
397:
53:
4417:, Annals of Mathematics Studies, vol. 76, Princeton, New Jersey: Princeton University Press,
4015:
Over a noncompact but paracompact base, this remains true provided one uses infinite
Grassmannian.
1466:
102:
4456:
3466:
2381:
385:. The tautological line bundle and the hyperplane bundle are exactly the two generators of the
3955:
2105:
1645:
436:
is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the
4422:
4390:
4354:
4326:
4302:
4213:
2078:
2051:
1518:
794:
763:
707:
644:
609:
502:
186:
734:
674:
185:
for vector bundles. Because of this, the tautological bundle is important in the study of
8:
3404:
2674:
841:
3971:
3709:
871:
821:
589:
569:
549:
529:
482:
462:
441:
148:
128:
82:
62:
35:
3792:= 1, the real tautological line bundle is none other than the well-known bundle whose
4434:
4394:
4376:
4340:
4309:
4288:
3728:
2042:
788:
416:
182:
3777:
4362:
4330:
3996:
3857:
2298:
420:
193:
177:
166:
4418:
4386:
4372:
4350:
4298:
3840:
3075:{\displaystyle L=\mathbf {Spec} \left({\mathcal {O}}_{\mathbb {P} ^{n}}/I\right)}
453:
437:
56:
3940:) and the tautological bundle is its opposite: the generator of negative degree.
3797:
4284:
4276:
3960:
1456:
which is clearly a homeomorphism. Hence, the result is a vector bundle of rank
634:
393:
3901:
4450:
4024:
In literature and textbooks, they are both often called canonical generators.
3990:
2963:
1486:
25:
4398:
317:. The hyperplane bundle is the line bundle corresponding to the hyperplane (
4410:
3805:
3793:
3773:
3716:
2389:
1543:
948:
638:
457:
386:
29:
4335:
2638:{\displaystyle {\begin{cases}O(H)\simeq O(1)\\f\mapsto fx_{0}\end{cases}}}
1948:{\displaystyle {\begin{cases}f_{E}:X\to G_{n}\\x\mapsto E_{x}\end{cases}}}
4406:
3965:
3950:
3853:
3720:
2291:
865:
405:
258:
17:
3708:
Over a field, its dual line bundle is the line bundle associated to the
3976:
3357:
701:
633:
are going to intersect. Fixing this up is a routine application of the
868:
is one tautological bundle, and the other, just described, is of rank
3852:
In the case of projective space, where the tautological bundle is a
2534:{\displaystyle \Gamma (U,O(D))=\{f\in K|(f)+D\geq 0{\text{ on }}U\}}
2297:
In algebraic geometry, the hyperplane bundle is the line bundle (as
1195:{\displaystyle G_{n}(\mathbb {R} ^{n+k})\times \mathbb {R} ^{n+k}.}
1107:
as follows. The total space of the bundle is the set of all pairs (
3458:{\displaystyle \mathbf {Spec} (\operatorname {Sym} {\check {E}})}
2662:
404:. The sphere bundle of the standard bundle is usually called the
2655:
is, as usual, viewed as a global section of the twisting sheaf
1672:
It is a universal bundle in the sense: for each compact space
3293:{\displaystyle \mathbb {A} ^{n+1}\times _{k}\mathbb {P} ^{n}}
1127:; it is given the subspace topology of the Cartesian product
2356:{\displaystyle H=\mathbb {P} ^{n-1}\subset \mathbb {P} ^{n}}
2031:{\displaystyle =\operatorname {Vect} _{n}^{\mathbb {R} }(X)}
668:, that now do not intersect. With this, we have the bundle.
4194:
2631:
1941:
1856:{\displaystyle E\hookrightarrow X\times \mathbb {R} ^{n+k}}
1787:
1463:
The above definition continues to make sense if we replace
1439:
2283:.) The rest is exactly like the tautological line bundle.
2667:
In algebraic geometry, this notion exists over any field
452:
Grassmannians by definition are the parameter spaces for
415:
More generally, there are also tautological bundles on a
2766:{\displaystyle \mathbb {P} ^{n}=\operatorname {Proj} A}
79:, given a point in the Grassmannian corresponding to a
1222:
under π, it is given a structure of a vector space by
247:{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1),}
4216:
4095:
4042:
3909:
3900:
the dual vector bundle) of the hyperplane bundle or
3866:
3814:
3737:
3694:{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1)}
3655:
3482:
3416:
3366:
3345:
is the tautological line bundle as defined before if
3314:
3249:
3208:
3195:{\displaystyle \mathbb {A} _{\mathbb {P} ^{n}}^{n+1}}
3159:
3095:
2975:
2782:
2738:
2677:
2569:
2451:
2398:
2313:
2226:
2138:
2108:
2081:
2054:
1973:
1880:
1822:
1685:
1648:
1603:
1552:
1521:
1494:
1469:
1295:
1133:
1062:
1003:
961:
902:
874:
844:
824:
797:
787:
that are their kernels, when considered as (rays of)
766:
737:
710:
677:
671:
The projective space case is included. By convention
647:
612:
592:
572:
552:
532:
505:
485:
465:
362:
327:
274:
205:
151:
131:
105:
85:
65:
38:
32:
310:{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(1)}
1258:). Finally, to see local triviality, given a point
396:'s "K-theory", the tautological line bundle over a
4229:
4200:
4079:
3928:
3888:
3831:
3748:
3693:
3634:
3473:of finite rank. Since we have the exact sequence:
3457:
3387:
3329:
3292:
3223:
3194:
3141:
3074:
2947:
2765:
2724:
2637:
2533:
2422:
2355:
2247:
2178:
2124:
2094:
2067:
2030:
1947:
1855:
1793:
1664:
1618:
1589:
1534:
1505:
1477:
1445:
1194:
1099:
1027:
985:
939:
880:
856:
830:
810:
779:
752:
723:
700:may usefully carry the tautological bundle in the
692:
660:
625:
598:
578:
558:
538:
518:
491:
471:
377:
348:
309:
246:
157:
137:
117:
91:
71:
44:
3788:In fact, it is straightforward to show that, for
2204:-space is defined as follows. The total space of
4448:
3089:is the ideal sheaf generated by global sections
637:device, so that the bundle projection is from a
4308:
430:has dropped out of favour, on the grounds that
4405:
4259:
2663:Tautological line bundle in algebraic geometry
2671:. The concrete definition is as follows. Let
1808:. The inverse map is given as follows: since
3649:, as defined above, corresponds to the dual
2528:
2482:
3800:. For a full proof of the above fact, see.
3407:. (cf. Hartshorne, Ch. I, the end of § 4.)
175:The tautological bundle is also called the
4361:
2430:one defines the corresponding line bundle
2179:{\displaystyle G_{n}(\mathbb {R} ^{n+k}).}
4334:
4283:, Advanced Book Classics (2nd ed.),
4152:
4119:
4080:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
4058:
3816:
3784:≥ 1. This remains true over other fields.
3739:
3667:
3602:
3506:
3369:
3349:is the field of real or complex numbers.
3317:
3280:
3252:
3211:
3169:
3162:
3012:
2934:
2905:
2877:
2870:
2813:
2741:
2407:
2343:
2322:
2229:
2154:
2010:
1837:
1730:
1590:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
1568:
1515:By definition, the infinite Grassmannian
1496:
1471:
1362:
1173:
1149:
1100:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
1078:
1006:
964:
940:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
918:
365:
330:
286:
217:
4431:Algebraic Geometry: A Concise Dictionary
2102:that corresponds to the identity map on
1626:Taking the direct limit of the bundles γ
169:the tautological bundle is known as the
447:
4449:
4275:
3979:(Thom spaces of tautological bundles γ
3469:corresponding to a locally free sheaf
2548:is the field of rational functions on
4428:
3142:{\displaystyle x_{i}y_{j}-x_{j}y_{i}}
2263:. The projection map π is given by π(
4433:, Berlin/Boston: Walter De Gruyter,
4325:, Wiley Classics Library, New York:
4241:, is a homeomorphism onto the image.
2188:
1816:is a subbundle of a trivial bundle:
1270:such that the orthogonal projection
891:
2423:{\displaystyle X=\mathbb {P} ^{n},}
1202:The projection map π is given by π(
1046:We define the tautological bundle γ
1028:{\displaystyle \mathbb {R} ^{n+k}.}
986:{\displaystyle \mathbb {R} ^{n+k};}
641:made up of identical copies of the
456:, of a given dimension, in a given
13:
4237:is the orthogonal projection onto
3912:
3889:{\displaystyle {\mathcal {O}}(-1)}
3869:
3659:
3594:
3498:
3388:{\displaystyle \mathbb {A} ^{n+1}}
3360:of the origin of the affine space
3004:
2805:
2452:
2384:. This can be seen as follows. If
2248:{\displaystyle \mathbb {R} ^{k+1}}
1610:
349:{\displaystyle \mathbb {P} ^{n-1}}
278:
209:
14:
4468:
3929:{\displaystyle {\mathcal {O}}(1)}
3727:. This is an example of an anti-
3231:; moreover, the closed points of
2294:of the tautological line bundle.
1119:of the Grassmannian and a vector
997:-dimensional vector subspaces of
955:-dimensional vector subspaces in
4319:Principles of algebraic geometry
3715:, whose global sections are the
3427:
3424:
3421:
3418:
3330:{\displaystyle \mathbb {P} ^{n}}
3224:{\displaystyle \mathbb {P} ^{n}}
2992:
2989:
2986:
2983:
2793:
2790:
2787:
2784:
419:of a vector bundle as well as a
378:{\displaystyle \mathbb {P} ^{n}}
3832:{\displaystyle \mathbb {P} (V)}
1676:, there is a natural bijection
1636:gives the tautological bundle γ
1039:= 1, it is the real projective
4253:
4244:
4178:
4168:
4147:
4138:
4135:
4114:
4087:is given a topology such that
4074:
4053:
4027:
4018:
4009:
3923:
3917:
3883:
3874:
3826:
3820:
3765:The tautological line bundle γ
3688:
3679:
3623:
3620:
3614:
3569:
3555:
3550:
3518:
3492:
3486:
3452:
3446:
3431:
3056:
3024:
2857:
2825:
2719:
2687:
2612:
2602:
2596:
2587:
2581:
2505:
2499:
2495:
2476:
2473:
2467:
2455:
2170:
2149:
2025:
2019:
1993:
1974:
1925:
1905:
1826:
1812:is compact, any vector bundle
1781:
1768:
1755:
1745:
1739:
1716:
1713:
1694:
1607:
1584:
1563:
1433:
1430:
1424:
1412:
1406:
1394:
1378:
1357:
1332:
1329:
1323:
1165:
1144:
1094:
1073:
993:as a set it is the set of all
934:
913:
760:carry the vector subspaces of
747:
741:
687:
681:
304:
298:
238:
229:
1:
4002:
3749:{\displaystyle \mathbb {C} ,}
3645:the tautological line bundle
1619:{\displaystyle k\to \infty .}
1506:{\displaystyle \mathbb {C} .}
99:-dimensional vector subspace
2279:is the dual vector space of
1478:{\displaystyle \mathbb {R} }
118:{\displaystyle W\subseteq V}
7:
3944:
444:could scarcely be avoided.
10:
4473:
4269:
4260:Milnor & Stasheff 1974
3202:over the same base scheme
731:the dual space, points of
3153:is a closed subscheme of
2208:is the set of all pairs (
1262:in the Grassmannian, let
389:of the projective space.
171:tautological line bundle.
3845:tautological line bundle
3759:
3304:is zero or the image of
2275:(so that the fiber over
1115:) consisting of a point
398:complex projective space
4277:Atiyah, Michael Francis
3467:algebraic vector bundle
3352:In more concise terms,
2382:homogeneous coordinates
2301:) corresponding to the
2259:a linear functional on
2216:) consisting of a line
1958:unique up to homotopy.
499:is a Grassmannian, and
165:itself. In the case of
4415:Characteristic Classes
4231:
4202:
4081:
3930:
3896:, the tensor inverse (
3890:
3833:
3750:
3703:Serre's twisting sheaf
3695:
3636:
3459:
3389:
3331:
3294:
3225:
3196:
3143:
3076:
2949:
2767:
2726:
2639:
2535:
2424:
2357:
2249:
2220:through the origin in
2180:
2126:
2125:{\displaystyle G_{n}.}
2096:
2069:
2032:
1949:
1857:
1795:
1666:
1665:{\displaystyle G_{n}.}
1620:
1591:
1536:
1507:
1479:
1447:
1196:
1101:
1029:
987:
941:
882:
858:
832:
812:
781:
754:
725:
694:
662:
627:
600:
580:
560:
540:
520:
493:
473:
379:
350:
311:
267:Serre's twisting sheaf
248:
187:characteristic classes
159:
139:
119:
93:
73:
46:
4429:Rubei, Elena (2014),
4336:10.1002/9781118032527
4327:John Wiley & Sons
4232:
4230:{\displaystyle p_{V}}
4203:
4082:
3956:Stiefel-Whitney class
3931:
3891:
3834:
3751:
3696:
3637:
3460:
3390:
3332:
3295:
3226:
3197:
3144:
3077:
2950:
2773:. Note that we have:
2768:
2727:
2640:
2536:
2425:
2358:
2250:
2200:on a real projective
2181:
2127:
2097:
2095:{\displaystyle G_{n}}
2070:
2068:{\displaystyle G_{n}}
2033:
1950:
1858:
1796:
1667:
1621:
1592:
1537:
1535:{\displaystyle G_{n}}
1508:
1480:
1448:
1197:
1102:
1030:
988:
942:
883:
859:
833:
813:
811:{\displaystyle V^{*}}
782:
780:{\displaystyle V^{*}}
755:
726:
724:{\displaystyle V^{*}}
704:sense. That is, with
695:
663:
661:{\displaystyle V_{g}}
628:
626:{\displaystyle V_{g}}
601:
581:
561:
541:
521:
519:{\displaystyle V_{g}}
494:
474:
380:
351:
312:
249:
160:
140:
120:
94:
74:
47:
4371:, Berlin, New York:
4214:
4093:
4040:
3907:
3864:
3812:
3735:
3653:
3480:
3414:
3364:
3312:
3247:
3206:
3157:
3093:
2973:
2780:
2736:
2675:
2567:
2449:
2396:
2311:
2224:
2136:
2106:
2079:
2052:
1971:
1878:
1820:
1683:
1646:
1601:
1550:
1519:
1492:
1467:
1293:
1282:isomorphically onto
1218:is the pre-image of
1131:
1060:
1001:
959:
900:
872:
842:
822:
795:
764:
753:{\displaystyle P(V)}
735:
708:
693:{\displaystyle P(V)}
675:
645:
610:
590:
570:
550:
530:
503:
483:
463:
448:Intuitive definition
402:standard line bundle
360:
325:
272:
203:
149:
129:
103:
83:
63:
36:
3808:of line bundles on
3405:exceptional divisor
3235:are exactly those (
3191:
2899:
2725:{\displaystyle A=k}
2015:
1735:
864:, the tautological
857:{\displaystyle n+1}
526:is the subspace of
22:tautological bundle
4411:Stasheff, James D.
4368:Algebraic Geometry
4310:Griffiths, Phillip
4262:, §2. Theorem 2.1.
4227:
4198:
4193:
4077:
3926:
3886:
3829:
3746:
3710:hyperplane divisor
3691:
3632:
3455:
3395:, where the locus
3385:
3327:
3290:
3221:
3192:
3160:
3139:
3072:
2945:
2868:
2763:
2722:
2635:
2630:
2531:
2420:
2353:
2303:hyperplane divisor
2245:
2176:
2122:
2092:
2065:
2028:
1999:
1945:
1940:
1871:determines a map
1853:
1791:
1786:
1719:
1662:
1616:
1587:
1532:
1503:
1475:
1443:
1438:
1286:, and then define
1266:be the set of all
1192:
1097:
1025:
983:
937:
878:
854:
828:
808:
789:linear functionals
777:
750:
721:
690:
658:
623:
596:
576:
556:
536:
516:
489:
469:
442:algebraic geometry
375:
346:
307:
244:
155:
135:
115:
89:
69:
42:
4440:978-3-11-031622-3
4382:978-0-387-90244-9
4363:Hartshorne, Robin
4346:978-0-471-05059-9
4294:978-0-201-09394-0
3989:→∞ is called the
3902:Serre twist sheaf
3856:, the associated
3729:ample line bundle
3583:
3449:
3300:such that either
2523:
2195:hyperplane bundle
2189:Hyperplane bundle
2043:paracompact space
892:Formal definition
881:{\displaystyle n}
831:{\displaystyle V}
599:{\displaystyle g}
579:{\displaystyle G}
559:{\displaystyle g}
546:corresponding to
539:{\displaystyle W}
492:{\displaystyle G}
472:{\displaystyle W}
417:projective bundle
263:hyperplane bundle
183:classifying space
158:{\displaystyle W}
138:{\displaystyle W}
125:, the fiber over
92:{\displaystyle k}
72:{\displaystyle V}
45:{\displaystyle k}
28:occurring over a
4464:
4443:
4425:
4401:
4357:
4338:
4324:
4305:
4263:
4257:
4251:
4248:
4242:
4236:
4234:
4233:
4228:
4226:
4225:
4207:
4205:
4204:
4199:
4197:
4196:
4190:
4189:
4167:
4166:
4155:
4134:
4133:
4122:
4113:
4112:
4086:
4084:
4083:
4078:
4073:
4072:
4061:
4052:
4051:
4031:
4025:
4022:
4016:
4013:
3997:Grassmann bundle
3935:
3933:
3932:
3927:
3916:
3915:
3895:
3893:
3892:
3887:
3873:
3872:
3858:invertible sheaf
3838:
3836:
3835:
3830:
3819:
3755:
3753:
3752:
3747:
3742:
3700:
3698:
3697:
3692:
3678:
3677:
3676:
3675:
3670:
3663:
3662:
3641:
3639:
3638:
3633:
3613:
3612:
3611:
3610:
3605:
3598:
3597:
3584:
3582:
3581:
3580:
3568:
3567:
3554:
3549:
3548:
3530:
3529:
3517:
3516:
3515:
3514:
3509:
3502:
3501:
3464:
3462:
3461:
3456:
3451:
3450:
3442:
3430:
3394:
3392:
3391:
3386:
3384:
3383:
3372:
3336:
3334:
3333:
3328:
3326:
3325:
3320:
3299:
3297:
3296:
3291:
3289:
3288:
3283:
3277:
3276:
3267:
3266:
3255:
3230:
3228:
3227:
3222:
3220:
3219:
3214:
3201:
3199:
3198:
3193:
3190:
3179:
3178:
3177:
3172:
3165:
3148:
3146:
3145:
3140:
3138:
3137:
3128:
3127:
3115:
3114:
3105:
3104:
3081:
3079:
3078:
3073:
3071:
3067:
3063:
3055:
3054:
3036:
3035:
3023:
3022:
3021:
3020:
3015:
3008:
3007:
2995:
2954:
2952:
2951:
2946:
2944:
2943:
2942:
2937:
2930:
2929:
2920:
2919:
2908:
2898:
2887:
2886:
2885:
2880:
2873:
2864:
2860:
2856:
2855:
2837:
2836:
2824:
2823:
2822:
2821:
2816:
2809:
2808:
2796:
2772:
2770:
2769:
2764:
2750:
2749:
2744:
2731:
2729:
2728:
2723:
2718:
2717:
2699:
2698:
2644:
2642:
2641:
2636:
2634:
2633:
2627:
2626:
2540:
2538:
2537:
2532:
2524:
2521:
2498:
2429:
2427:
2426:
2421:
2416:
2415:
2410:
2362:
2360:
2359:
2354:
2352:
2351:
2346:
2337:
2336:
2325:
2299:invertible sheaf
2286:In other words,
2254:
2252:
2251:
2246:
2244:
2243:
2232:
2185:
2183:
2182:
2177:
2169:
2168:
2157:
2148:
2147:
2131:
2129:
2128:
2123:
2118:
2117:
2101:
2099:
2098:
2093:
2091:
2090:
2074:
2072:
2071:
2066:
2064:
2063:
2037:
2035:
2034:
2029:
2014:
2013:
2007:
1992:
1991:
1954:
1952:
1951:
1946:
1944:
1943:
1937:
1936:
1917:
1916:
1898:
1897:
1862:
1860:
1859:
1854:
1852:
1851:
1840:
1800:
1798:
1797:
1792:
1790:
1789:
1780:
1779:
1767:
1766:
1734:
1733:
1727:
1712:
1711:
1671:
1669:
1668:
1663:
1658:
1657:
1625:
1623:
1622:
1617:
1596:
1594:
1593:
1588:
1583:
1582:
1571:
1562:
1561:
1541:
1539:
1538:
1533:
1531:
1530:
1512:
1510:
1509:
1504:
1499:
1484:
1482:
1481:
1476:
1474:
1452:
1450:
1449:
1444:
1442:
1441:
1377:
1376:
1365:
1356:
1355:
1322:
1321:
1201:
1199:
1198:
1193:
1188:
1187:
1176:
1164:
1163:
1152:
1143:
1142:
1106:
1104:
1103:
1098:
1093:
1092:
1081:
1072:
1071:
1035:For example, if
1034:
1032:
1031:
1026:
1021:
1020:
1009:
992:
990:
989:
984:
979:
978:
967:
946:
944:
943:
938:
933:
932:
921:
912:
911:
887:
885:
884:
879:
863:
861:
860:
855:
837:
835:
834:
829:
817:
815:
814:
809:
807:
806:
786:
784:
783:
778:
776:
775:
759:
757:
756:
751:
730:
728:
727:
722:
720:
719:
699:
697:
696:
691:
667:
665:
664:
659:
657:
656:
632:
630:
629:
624:
622:
621:
605:
603:
602:
597:
585:
583:
582:
577:
565:
563:
562:
557:
545:
543:
542:
537:
525:
523:
522:
517:
515:
514:
498:
496:
495:
490:
478:
476:
475:
470:
454:linear subspaces
428:canonical bundle
421:Grassmann bundle
384:
382:
381:
376:
374:
373:
368:
355:
353:
352:
347:
345:
344:
333:
316:
314:
313:
308:
297:
296:
295:
294:
289:
282:
281:
253:
251:
250:
245:
228:
227:
226:
225:
220:
213:
212:
194:invertible sheaf
178:universal bundle
167:projective space
164:
162:
161:
156:
145:is the subspace
144:
142:
141:
136:
124:
122:
121:
116:
98:
96:
95:
90:
78:
76:
75:
70:
51:
49:
48:
43:
4472:
4471:
4467:
4466:
4465:
4463:
4462:
4461:
4447:
4446:
4441:
4407:Milnor, John W.
4383:
4373:Springer-Verlag
4347:
4322:
4295:
4272:
4267:
4266:
4258:
4254:
4249:
4245:
4221:
4217:
4215:
4212:
4211:
4192:
4191:
4185:
4181:
4172:
4171:
4156:
4151:
4150:
4123:
4118:
4117:
4108:
4104:
4097:
4096:
4094:
4091:
4090:
4062:
4057:
4056:
4047:
4043:
4041:
4038:
4037:
4032:
4028:
4023:
4019:
4014:
4010:
4005:
3984:
3972:Borel's theorem
3947:
3911:
3910:
3908:
3905:
3904:
3868:
3867:
3865:
3862:
3861:
3860:of sections is
3847:is a generator.
3841:infinite cyclic
3815:
3813:
3810:
3809:
3774:locally trivial
3771:
3762:
3738:
3736:
3733:
3732:
3671:
3666:
3665:
3664:
3658:
3657:
3656:
3654:
3651:
3650:
3606:
3601:
3600:
3599:
3593:
3592:
3591:
3576:
3572:
3563:
3559:
3558:
3553:
3544:
3540:
3525:
3521:
3510:
3505:
3504:
3503:
3497:
3496:
3495:
3481:
3478:
3477:
3441:
3440:
3417:
3415:
3412:
3411:
3373:
3368:
3367:
3365:
3362:
3361:
3321:
3316:
3315:
3313:
3310:
3309:
3284:
3279:
3278:
3272:
3268:
3256:
3251:
3250:
3248:
3245:
3244:
3215:
3210:
3209:
3207:
3204:
3203:
3180:
3173:
3168:
3167:
3166:
3161:
3158:
3155:
3154:
3133:
3129:
3123:
3119:
3110:
3106:
3100:
3096:
3094:
3091:
3090:
3059:
3050:
3046:
3031:
3027:
3016:
3011:
3010:
3009:
3003:
3002:
3001:
3000:
2996:
2982:
2974:
2971:
2970:
2938:
2933:
2932:
2931:
2925:
2921:
2909:
2904:
2903:
2888:
2881:
2876:
2875:
2874:
2869:
2851:
2847:
2832:
2828:
2817:
2812:
2811:
2810:
2804:
2803:
2802:
2801:
2797:
2783:
2781:
2778:
2777:
2745:
2740:
2739:
2737:
2734:
2733:
2713:
2709:
2694:
2690:
2676:
2673:
2672:
2665:
2654:
2629:
2628:
2622:
2618:
2606:
2605:
2571:
2570:
2568:
2565:
2564:
2520:
2494:
2450:
2447:
2446:
2411:
2406:
2405:
2397:
2394:
2393:
2378:
2372:
2366:given as, say,
2347:
2342:
2341:
2326:
2321:
2320:
2312:
2309:
2308:
2233:
2228:
2227:
2225:
2222:
2221:
2191:
2158:
2153:
2152:
2143:
2139:
2137:
2134:
2133:
2113:
2109:
2107:
2104:
2103:
2086:
2082:
2080:
2077:
2076:
2059:
2055:
2053:
2050:
2049:
2009:
2008:
2003:
1987:
1983:
1972:
1969:
1968:
1939:
1938:
1932:
1928:
1919:
1918:
1912:
1908:
1893:
1889:
1882:
1881:
1879:
1876:
1875:
1841:
1836:
1835:
1821:
1818:
1817:
1785:
1784:
1775:
1771:
1762:
1758:
1749:
1748:
1729:
1728:
1723:
1707:
1703:
1687:
1686:
1684:
1681:
1680:
1653:
1649:
1647:
1644:
1643:
1641:
1635:
1602:
1599:
1598:
1572:
1567:
1566:
1557:
1553:
1551:
1548:
1547:
1526:
1522:
1520:
1517:
1516:
1495:
1493:
1490:
1489:
1470:
1468:
1465:
1464:
1437:
1436:
1388:
1387:
1366:
1361:
1360:
1351:
1347:
1314:
1310:
1297:
1296:
1294:
1291:
1290:
1177:
1172:
1171:
1153:
1148:
1147:
1138:
1134:
1132:
1129:
1128:
1082:
1077:
1076:
1067:
1063:
1061:
1058:
1057:
1055:
1010:
1005:
1004:
1002:
999:
998:
968:
963:
962:
960:
957:
956:
922:
917:
916:
907:
903:
901:
898:
897:
894:
873:
870:
869:
843:
840:
839:
823:
820:
819:
802:
798:
796:
793:
792:
771:
767:
765:
762:
761:
736:
733:
732:
715:
711:
709:
706:
705:
676:
673:
672:
652:
648:
646:
643:
642:
617:
613:
611:
608:
607:
591:
588:
587:
571:
568:
567:
551:
548:
547:
531:
528:
527:
510:
506:
504:
501:
500:
484:
481:
480:
464:
461:
460:
450:
438:canonical class
426:The older term
369:
364:
363:
361:
358:
357:
334:
329:
328:
326:
323:
322:
290:
285:
284:
283:
277:
276:
275:
273:
270:
269:
221:
216:
215:
214:
208:
207:
206:
204:
201:
200:
150:
147:
146:
130:
127:
126:
104:
101:
100:
84:
81:
80:
64:
61:
60:
37:
34:
33:
12:
11:
5:
4470:
4460:
4459:
4457:Vector bundles
4445:
4444:
4439:
4426:
4403:
4381:
4359:
4345:
4314:Harris, Joseph
4306:
4293:
4285:Addison-Wesley
4271:
4268:
4265:
4264:
4252:
4243:
4224:
4220:
4209:
4208:
4195:
4188:
4184:
4180:
4177:
4174:
4173:
4170:
4165:
4162:
4159:
4154:
4149:
4146:
4143:
4140:
4137:
4132:
4129:
4126:
4121:
4116:
4111:
4107:
4103:
4102:
4100:
4076:
4071:
4068:
4065:
4060:
4055:
4050:
4046:
4036:is open since
4026:
4017:
4007:
4006:
4004:
4001:
4000:
3999:
3994:
3980:
3974:
3969:
3963:
3961:Euler sequence
3958:
3953:
3946:
3943:
3942:
3941:
3925:
3922:
3919:
3914:
3885:
3882:
3879:
3876:
3871:
3849:
3848:
3828:
3825:
3822:
3818:
3786:
3785:
3766:
3761:
3758:
3745:
3741:
3690:
3687:
3684:
3681:
3674:
3669:
3661:
3643:
3642:
3631:
3628:
3625:
3622:
3619:
3616:
3609:
3604:
3596:
3590:
3587:
3579:
3575:
3571:
3566:
3562:
3557:
3552:
3547:
3543:
3539:
3536:
3533:
3528:
3524:
3520:
3513:
3508:
3500:
3494:
3491:
3488:
3485:
3454:
3448:
3445:
3439:
3436:
3433:
3429:
3426:
3423:
3420:
3382:
3379:
3376:
3371:
3324:
3319:
3287:
3282:
3275:
3271:
3265:
3262:
3259:
3254:
3218:
3213:
3189:
3186:
3183:
3176:
3171:
3164:
3136:
3132:
3126:
3122:
3118:
3113:
3109:
3103:
3099:
3083:
3082:
3070:
3066:
3062:
3058:
3053:
3049:
3045:
3042:
3039:
3034:
3030:
3026:
3019:
3014:
3006:
2999:
2994:
2991:
2988:
2985:
2981:
2978:
2956:
2955:
2941:
2936:
2928:
2924:
2918:
2915:
2912:
2907:
2902:
2897:
2894:
2891:
2884:
2879:
2872:
2867:
2863:
2859:
2854:
2850:
2846:
2843:
2840:
2835:
2831:
2827:
2820:
2815:
2807:
2800:
2795:
2792:
2789:
2786:
2762:
2759:
2756:
2753:
2748:
2743:
2721:
2716:
2712:
2708:
2705:
2702:
2697:
2693:
2689:
2686:
2683:
2680:
2664:
2661:
2652:
2646:
2645:
2632:
2625:
2621:
2617:
2614:
2611:
2608:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2576:
2574:
2542:
2541:
2530:
2527:
2522: on
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2497:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2419:
2414:
2409:
2404:
2401:
2390:(Weil) divisor
2376:
2370:
2364:
2363:
2350:
2345:
2340:
2335:
2332:
2329:
2324:
2319:
2316:
2242:
2239:
2236:
2231:
2190:
2187:
2175:
2172:
2167:
2164:
2161:
2156:
2151:
2146:
2142:
2121:
2116:
2112:
2089:
2085:
2062:
2058:
2039:
2038:
2027:
2024:
2021:
2018:
2012:
2006:
2002:
1998:
1995:
1990:
1986:
1982:
1979:
1976:
1956:
1955:
1942:
1935:
1931:
1927:
1924:
1921:
1920:
1915:
1911:
1907:
1904:
1901:
1896:
1892:
1888:
1887:
1885:
1850:
1847:
1844:
1839:
1834:
1831:
1828:
1825:
1802:
1801:
1788:
1783:
1778:
1774:
1770:
1765:
1761:
1757:
1754:
1751:
1750:
1747:
1744:
1741:
1738:
1732:
1726:
1722:
1718:
1715:
1710:
1706:
1702:
1699:
1696:
1693:
1692:
1690:
1661:
1656:
1652:
1637:
1627:
1615:
1612:
1609:
1606:
1586:
1581:
1578:
1575:
1570:
1565:
1560:
1556:
1529:
1525:
1502:
1498:
1473:
1454:
1453:
1440:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1389:
1386:
1383:
1380:
1375:
1372:
1369:
1364:
1359:
1354:
1350:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1320:
1317:
1313:
1309:
1306:
1303:
1302:
1300:
1191:
1186:
1183:
1180:
1175:
1170:
1167:
1162:
1159:
1156:
1151:
1146:
1141:
1137:
1096:
1091:
1088:
1085:
1080:
1075:
1070:
1066:
1047:
1024:
1019:
1016:
1013:
1008:
982:
977:
974:
971:
966:
936:
931:
928:
925:
920:
915:
910:
906:
893:
890:
877:
853:
850:
847:
838:has dimension
827:
805:
801:
774:
770:
749:
746:
743:
740:
718:
714:
689:
686:
683:
680:
655:
651:
635:disjoint union
620:
616:
595:
575:
555:
535:
513:
509:
488:
468:
449:
446:
410:Bott generator
400:is called the
394:Michael Atiyah
372:
367:
343:
340:
337:
332:
306:
303:
300:
293:
288:
280:
255:
254:
243:
240:
237:
234:
231:
224:
219:
211:
154:
134:
114:
111:
108:
88:
68:
41:
9:
6:
4:
3:
2:
4469:
4458:
4455:
4454:
4452:
4442:
4436:
4432:
4427:
4424:
4420:
4416:
4412:
4408:
4404:
4400:
4396:
4392:
4388:
4384:
4378:
4374:
4370:
4369:
4364:
4360:
4356:
4352:
4348:
4342:
4337:
4332:
4328:
4321:
4320:
4315:
4311:
4307:
4304:
4300:
4296:
4290:
4286:
4282:
4278:
4274:
4273:
4261:
4256:
4247:
4240:
4222:
4218:
4186:
4182:
4175:
4163:
4160:
4157:
4144:
4141:
4130:
4127:
4124:
4109:
4105:
4098:
4089:
4088:
4069:
4066:
4063:
4048:
4044:
4035:
4030:
4021:
4012:
4008:
3998:
3995:
3992:
3991:Thom spectrum
3988:
3983:
3978:
3975:
3973:
3970:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3948:
3939:
3920:
3903:
3899:
3880:
3877:
3859:
3855:
3851:
3850:
3846:
3842:
3823:
3807:
3803:
3802:
3801:
3799:
3795:
3791:
3783:
3779:
3775:
3770:
3764:
3763:
3757:
3743:
3730:
3726:
3722:
3718:
3714:
3711:
3706:
3704:
3685:
3682:
3672:
3648:
3629:
3626:
3617:
3607:
3588:
3585:
3577:
3573:
3564:
3560:
3545:
3541:
3537:
3534:
3531:
3526:
3522:
3511:
3489:
3483:
3476:
3475:
3474:
3472:
3468:
3443:
3437:
3434:
3408:
3406:
3402:
3398:
3380:
3377:
3374:
3359:
3355:
3350:
3348:
3344:
3340:
3322:
3307:
3303:
3285:
3273:
3269:
3263:
3260:
3257:
3242:
3238:
3234:
3216:
3187:
3184:
3181:
3174:
3152:
3134:
3130:
3124:
3120:
3116:
3111:
3107:
3101:
3097:
3088:
3068:
3064:
3060:
3051:
3047:
3043:
3040:
3037:
3032:
3028:
3017:
2997:
2979:
2976:
2969:
2968:
2967:
2965:
2964:relative Spec
2961:
2939:
2926:
2922:
2916:
2913:
2910:
2900:
2895:
2892:
2889:
2882:
2865:
2861:
2852:
2848:
2844:
2841:
2838:
2833:
2829:
2818:
2798:
2776:
2775:
2774:
2760:
2757:
2754:
2751:
2746:
2714:
2710:
2706:
2703:
2700:
2695:
2691:
2684:
2681:
2678:
2670:
2660:
2658:
2651:
2623:
2619:
2615:
2609:
2599:
2593:
2590:
2584:
2578:
2572:
2563:
2562:
2561:
2559:
2555:
2551:
2547:
2525:
2517:
2514:
2511:
2508:
2502:
2491:
2488:
2485:
2479:
2470:
2464:
2461:
2458:
2445:
2444:
2443:
2441:
2437:
2433:
2417:
2412:
2402:
2399:
2391:
2387:
2383:
2379:
2369:
2348:
2338:
2333:
2330:
2327:
2317:
2314:
2307:
2306:
2305:
2304:
2300:
2295:
2293:
2289:
2284:
2282:
2278:
2274:
2270:
2266:
2262:
2258:
2240:
2237:
2234:
2219:
2215:
2211:
2207:
2203:
2199:
2196:
2186:
2173:
2165:
2162:
2159:
2144:
2140:
2119:
2114:
2110:
2087:
2083:
2060:
2056:
2047:
2044:
2022:
2016:
2004:
2000:
1996:
1988:
1984:
1980:
1977:
1967:
1966:
1965:
1963:
1959:
1933:
1929:
1922:
1913:
1909:
1902:
1899:
1894:
1890:
1883:
1874:
1873:
1872:
1870:
1866:
1848:
1845:
1842:
1832:
1829:
1823:
1815:
1811:
1807:
1776:
1772:
1763:
1759:
1752:
1742:
1736:
1724:
1720:
1708:
1704:
1700:
1697:
1688:
1679:
1678:
1677:
1675:
1659:
1654:
1650:
1640:
1634:
1630:
1613:
1604:
1579:
1576:
1573:
1558:
1554:
1545:
1527:
1523:
1513:
1500:
1488:
1487:complex field
1461:
1459:
1427:
1421:
1418:
1415:
1409:
1403:
1400:
1397:
1391:
1384:
1381:
1373:
1370:
1367:
1352:
1348:
1344:
1341:
1338:
1335:
1326:
1318:
1315:
1311:
1307:
1304:
1298:
1289:
1288:
1287:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1205:
1189:
1184:
1181:
1178:
1168:
1160:
1157:
1154:
1139:
1135:
1126:
1122:
1118:
1114:
1110:
1089:
1086:
1083:
1068:
1064:
1054:
1050:
1044:
1042:
1038:
1022:
1017:
1014:
1011:
996:
980:
975:
972:
969:
954:
950:
929:
926:
923:
908:
904:
889:
875:
867:
851:
848:
845:
825:
803:
799:
790:
772:
768:
744:
738:
716:
712:
703:
684:
678:
669:
653:
649:
640:
636:
618:
614:
593:
573:
553:
533:
511:
507:
486:
466:
459:
455:
445:
443:
439:
435:
434:
429:
424:
422:
418:
413:
411:
407:
403:
399:
395:
390:
388:
370:
341:
338:
335:
320:
301:
291:
268:
264:
260:
241:
235:
232:
222:
199:
198:
197:
195:
190:
188:
184:
180:
179:
173:
172:
168:
152:
132:
112:
109:
106:
86:
66:
58:
55:
39:
31:
27:
26:vector bundle
23:
19:
4430:
4414:
4366:
4318:
4280:
4255:
4246:
4238:
4033:
4029:
4020:
4011:
3986:
3981:
3897:
3806:Picard group
3798:Möbius strip
3789:
3787:
3781:
3768:
3724:
3717:linear forms
3712:
3707:
3646:
3644:
3470:
3410:In general,
3409:
3400:
3396:
3353:
3351:
3346:
3342:
3338:
3305:
3301:
3240:
3236:
3232:
3150:
3086:
3084:
2966:. Now, put:
2959:
2957:
2668:
2666:
2656:
2649:
2647:
2557:
2553:
2549:
2545:
2543:
2439:
2435:
2431:
2385:
2374:
2367:
2365:
2302:
2296:
2287:
2285:
2280:
2276:
2272:
2268:
2264:
2260:
2256:
2217:
2213:
2209:
2205:
2201:
2197:
2194:
2192:
2045:
2040:
1961:
1960:
1957:
1868:
1864:
1813:
1809:
1805:
1803:
1673:
1638:
1632:
1628:
1544:direct limit
1514:
1462:
1457:
1455:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1124:
1120:
1116:
1112:
1108:
1052:
1048:
1045:
1040:
1036:
994:
952:
949:Grassmannian
895:
670:
458:vector space
451:
431:
427:
425:
414:
401:
391:
387:Picard group
262:
256:
191:
176:
174:
170:
30:Grassmannian
21:
15:
3966:Chern class
3951:Hopf bundle
3854:line bundle
3794:total space
3721:Chern class
2560:, we have:
2292:dual bundle
866:line bundle
639:total space
406:Hopf bundle
54:dimensional
18:mathematics
4003:References
3977:Thom space
3843:, and the
2373:= 0, when
702:dual space
4179:↦
4145:
4139:→
3878:−
3683:−
3624:→
3589:
3570:↦
3556:⟶
3535:…
3493:→
3487:→
3447:ˇ
3438:
3270:×
3117:−
3041:…
2923:×
2842:…
2758:
2704:…
2613:↦
2591:≃
2552:. Taking
2515:≥
2489:∈
2453:Γ
2339:⊂
2331:−
2017:
1926:↦
1906:→
1863:for some
1833:×
1827:↪
1773:γ
1764:∗
1756:↦
1737:
1717:→
1611:∞
1608:→
1485:with the
1392:ϕ
1382:×
1345:⊆
1339:×
1333:→
1316:−
1312:π
1305:ϕ
1169:×
804:∗
773:∗
717:∗
433:canonical
339:−
233:−
110:⊆
57:subspaces
4451:Category
4413:(1974),
4399:13348052
4365:(1977),
4316:(1994),
4281:K-theory
4279:(1989),
3945:See also
3776:but not
3341:. Thus,
2380:are the
2048:. Since
2041:for any
1043:-space.
4423:0440554
4391:0463157
4355:1288523
4303:1043170
4270:Sources
3938:divisor
3796:is the
3778:trivial
3731:. Over
3465:is the
3403:is the
3399:= 0 in
3358:blow-up
3356:is the
3149:. Then
2290:is the
1867:and so
1542:is the
947:be the
408:. (cf.
319:divisor
261:of the
4437:
4421:
4397:
4389:
4379:
4353:
4343:
4301:
4291:
4210:where
3780:, for
3719:. Its
3085:where
2958:where
2648:where
2556:to be
2544:where
1962:Remark
20:, the
4323:(PDF)
3760:Facts
3243:) of
2438:) on
2388:is a
1278:maps
1274:onto
1246:) = (
1214:. If
1056:over
818:. If
479:. If
196:) is
24:is a
4435:ISBN
4395:OCLC
4377:ISBN
4341:ISBN
4289:ISBN
3804:The
3772:is
3723:is −
2960:Spec
2755:Proj
2732:and
2442:by
2271:) =
2255:and
2193:The
2001:Vect
1721:Vect
1234:) +
1210:) =
896:Let
259:dual
257:the
4331:doi
4142:End
3985:as
3839:is
3767:1,
3701:of
3586:Sym
3435:Sym
3337:is
3308:in
2962:is
2392:on
1642:of
1597:as
1546:of
1123:in
951:of
791:on
566:in
440:in
412:.)
392:In
356:in
265:or
59:of
16:In
4453::
4419:MR
4409:;
4393:,
4387:MR
4385:,
4375:,
4351:MR
4349:,
4339:,
4329:,
4312:;
4299:MR
4297:,
4287:,
3993:.)
3898:ie
3239:,
2267:,
2212:,
1631:,
1460:.
1256:bw
1254:+
1252:av
1250:,
1242:,
1230:,
1206:,
1111:,
1051:,
888:.
423:.
321:)
189:.
4402:.
4358:.
4333::
4239:V
4223:V
4219:p
4187:V
4183:p
4176:V
4169:)
4164:k
4161:+
4158:n
4153:R
4148:(
4136:)
4131:k
4128:+
4125:n
4120:R
4115:(
4110:n
4106:G
4099:{
4075:)
4070:k
4067:+
4064:n
4059:R
4054:(
4049:n
4045:G
4034:U
3987:n
3982:n
3924:)
3921:1
3918:(
3913:O
3884:)
3881:1
3875:(
3870:O
3827:)
3824:V
3821:(
3817:P
3790:k
3782:k
3769:k
3744:,
3740:C
3725:H
3713:H
3689:)
3686:1
3680:(
3673:n
3668:P
3660:O
3647:L
3630:,
3627:0
3621:)
3618:1
3615:(
3608:n
3603:P
3595:O
3578:i
3574:y
3565:i
3561:x
3551:]
3546:n
3542:x
3538:,
3532:,
3527:0
3523:x
3519:[
3512:n
3507:P
3499:O
3490:I
3484:0
3471:E
3453:)
3444:E
3432:(
3428:c
3425:e
3422:p
3419:S
3401:L
3397:x
3381:1
3378:+
3375:n
3370:A
3354:L
3347:k
3343:L
3339:y
3323:n
3318:P
3306:x
3302:x
3286:n
3281:P
3274:k
3264:1
3261:+
3258:n
3253:A
3241:y
3237:x
3233:L
3217:n
3212:P
3188:1
3185:+
3182:n
3175:n
3170:P
3163:A
3151:L
3135:i
3131:y
3125:j
3121:x
3112:j
3108:y
3102:i
3098:x
3087:I
3069:)
3065:I
3061:/
3057:]
3052:n
3048:x
3044:,
3038:,
3033:0
3029:x
3025:[
3018:n
3013:P
3005:O
2998:(
2993:c
2990:e
2987:p
2984:S
2980:=
2977:L
2940:n
2935:P
2927:k
2917:1
2914:+
2911:n
2906:A
2901:=
2896:1
2893:+
2890:n
2883:n
2878:P
2871:A
2866:=
2862:)
2858:]
2853:n
2849:x
2845:,
2839:,
2834:0
2830:x
2826:[
2819:n
2814:P
2806:O
2799:(
2794:c
2791:e
2788:p
2785:S
2761:A
2752:=
2747:n
2742:P
2720:]
2715:n
2711:y
2707:,
2701:,
2696:0
2692:y
2688:[
2685:k
2682:=
2679:A
2669:k
2657:O
2653:0
2650:x
2624:0
2620:x
2616:f
2610:f
2603:)
2600:1
2597:(
2594:O
2588:)
2585:H
2582:(
2579:O
2573:{
2558:H
2554:D
2550:X
2546:K
2529:}
2526:U
2518:0
2512:D
2509:+
2506:)
2503:f
2500:(
2496:|
2492:K
2486:f
2483:{
2480:=
2477:)
2474:)
2471:D
2468:(
2465:O
2462:,
2459:U
2456:(
2440:X
2436:D
2434:(
2432:O
2418:,
2413:n
2408:P
2403:=
2400:X
2386:D
2377:i
2375:x
2371:0
2368:x
2349:n
2344:P
2334:1
2328:n
2323:P
2318:=
2315:H
2288:H
2281:L
2277:L
2273:L
2269:f
2265:L
2261:L
2257:f
2241:1
2238:+
2235:k
2230:R
2218:L
2214:f
2210:L
2206:H
2202:k
2198:H
2174:.
2171:)
2166:k
2163:+
2160:n
2155:R
2150:(
2145:n
2141:G
2120:.
2115:n
2111:G
2088:n
2084:G
2061:n
2057:G
2046:X
2026:)
2023:X
2020:(
2011:R
2005:n
1997:=
1994:]
1989:n
1985:G
1981:,
1978:X
1975:[
1934:x
1930:E
1923:x
1914:n
1910:G
1903:X
1900::
1895:E
1891:f
1884:{
1869:E
1865:k
1849:k
1846:+
1843:n
1838:R
1830:X
1824:E
1814:E
1810:X
1806:n
1782:)
1777:n
1769:(
1760:f
1753:f
1746:)
1743:X
1740:(
1731:R
1725:n
1714:]
1709:n
1705:G
1701:,
1698:X
1695:[
1689:{
1674:X
1660:.
1655:n
1651:G
1639:n
1633:k
1629:n
1614:.
1605:k
1585:)
1580:k
1577:+
1574:n
1569:R
1564:(
1559:n
1555:G
1528:n
1524:G
1501:.
1497:C
1472:R
1458:n
1434:)
1431:)
1428:v
1425:(
1422:p
1419:,
1416:V
1413:(
1410:=
1407:)
1404:v
1401:,
1398:V
1395:(
1385:X
1379:)
1374:k
1371:+
1368:n
1363:R
1358:(
1353:n
1349:G
1342:X
1336:U
1330:)
1327:U
1324:(
1319:1
1308::
1299:{
1284:X
1280:V
1276:X
1272:p
1268:V
1264:U
1260:X
1248:V
1244:w
1240:V
1238:(
1236:b
1232:v
1228:V
1226:(
1224:a
1220:V
1216:F
1212:V
1208:v
1204:V
1190:.
1185:k
1182:+
1179:n
1174:R
1166:)
1161:k
1158:+
1155:n
1150:R
1145:(
1140:n
1136:G
1125:V
1121:v
1117:V
1113:v
1109:V
1095:)
1090:k
1087:+
1084:n
1079:R
1074:(
1069:n
1065:G
1053:k
1049:n
1041:k
1037:n
1023:.
1018:k
1015:+
1012:n
1007:R
995:n
981:;
976:k
973:+
970:n
965:R
953:n
935:)
930:k
927:+
924:n
919:R
914:(
909:n
905:G
876:n
852:1
849:+
846:n
826:V
800:V
769:V
748:)
745:V
742:(
739:P
713:V
688:)
685:V
682:(
679:P
654:g
650:V
619:g
615:V
594:g
574:G
554:g
534:W
512:g
508:V
487:G
467:W
371:n
366:P
342:1
336:n
331:P
305:)
302:1
299:(
292:n
287:P
279:O
242:,
239:)
236:1
230:(
223:n
218:P
210:O
153:W
133:W
113:V
107:W
87:k
67:V
52:-
40:k
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