Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/talks/2009osaka-talk.pdf Äàòà èçìåíåíèÿ: Sun Apr 8 15:56:54 2012 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:03:34 2016 Êîäèðîâêà:

The Universal Toric Genus
Taras Panov Moscow State University

joint with Victor Buchstab er and Nigel Ray

The 36th Symp osium on Transformation Groups, Osaka, 10­12 Decemb er 2009


Motivations: non-equivariant cob ordism and torus actions. Thm [Buchstab er-P.-Ray]. Every complex cob ordism class in dim > 2 contains a quasitoric manifold.

In other words, every stably complex manifold is cob ordant to a manifold with a nicely b ehaving torus action. All manifolds are smo oth and closed, unless otherwise stated.
n n M1 M2 (co)b ordant if there is a manifold with b oundary W n+1 such that W n+1 = M1 M2.

2


Complex b ordism: work with complex manifolds. complex mflds almost complex mflds stably complex mflds Stably complex structure on a 2n-dim manifold M is determined by a choice of isomorphism
= 2(l-n) - c : M R

where is an l-dim complex vector bundle. Complex b ordism classes [M , c ] form the complex b ordism ring U U = (pt) with resp ect to the disjoint union and pro duct. U = Z[a1, a2, ...], dim ai = 2i Novikov'1960.

3


Quasitoric manifolds: 2n-dimensional manifolds M with a "nice" action of the torus T n (after Davis­Januszkiewicz);

· the T n-action is lo cally standard (lo cally lo oks like the standard T n-representation in Cn); · the orbit space M /T n is an n-dim simple p olytop e P .

Examples include projective smo oth toric varieties and symplectic manifolds M with Hamiltonian actions of T n (also known as toric manifolds). In their turn, quasitoric manifolds are examples of torus manifolds of Hattori­Masuda.

4


Equivariant cob ordism and the universal toric genus. X a T k -space. There are 3 equivariant complex cob ordism theories: ·

of stably tangentially complex T k -bundles over X (here X is a smo oth manifold ). · M U k (X ) = lim[S V X+, M UT k (W )]T k : homotopic T k -cob ordisms; T here M UT k (W ) is the Thom T k -space of the universal |W |dimensional complex T k -vector bundle |W |, and S V is the unit sphere in a T k -representation space V .
· U (E T k âT k X ): Borel T k -cob ordisms.

U :T

k

(X ): geometric T k -cob ordisms: set of cob ordism classes

5


There are natural transformations of cohomology theories
U :T k (X ) - M UT k (X ) - U (E T k âT k X ). Restricting to X = pt we get a map := · : U :T k - U (B T k ) = U [[u1, . . . , uk ]], which we refer to as the universal toric genus. It assigns to the cob ordism class [M , c ] -2nk of a 2n-dimensional T k -manifold M

the "cob ordism class" of the map E T k âT k M B T k . We may write (M , c ) =


U :T

g (M )u ,

U where = (1, . . . , k ) Nk , u = u11 · . . . · uk k , g (M ) 2(||+n). U We have g0(M ) = [M ] 2n. How to express the other co efficients g (M )?
6


U Ray's basis in (B T k ).

Consider the pro duct of unit 3-spheres (S 3)j =
{

(z1, . . . , z2j ) C2j : |zi|2 + |zi+j |2 = 1 for 1

i

j

}

with the free T j -action by (t1, . . . , tj )·(z1, . . . , z2j ) = (t-1z1, t-1t-1z2, . . . , t-11t-1zj , t1zj +1, . . . , tj z2j ) 1 12 j- j Bj := (S 3)j /T j : b ounded flag manifold. It is a Bott manifold, i.e. the total space of a j -fold iterated S 2-bundle over B0 = . For 1 i j there are complex line bundles

i : (S 3)j âT j C - Bj via the action (t1, . . . , tj ) · z = tiz for z C. For any j > 0 have an explicit isomorphism ¯ (Bj ) Cj = 1 12 · · · j -1j 1 · · · which defines a stably cplx structure c on Bj with [Bj , c ] j j

¯ j ,
U = 0 in 2j .
7


U Prop 1. The basis element b 2||(B T k ) dual to u U (B T k ) is represented geometrically by the classifying map

: B - B T k for the pro duct 1â . . . â k of line bundles over B = B1 â . . . â Bk . Let T = T 1 â . . . â T k ,and (S 3) = (S 3)1 â . . . â (S 3)k , on which T acts co ordinatewise. Define G (M ) := (S 3) âT M , where T acts on M via the representation
1 (t1,1, . . . , t1,1 ; . . . ; tk,1, . . . , tk,k ) - (t-, , . . . , t-1 ). 11 k,k

Thm 2. The manifold G (M ) represents the co efficient g (M ) U 2(||+n) of the universal toric genus expansion.
8


Hirzebruch genera and equivariant extentions. R a (graded) commutative ring with unit.
U : R a Hirzebruch genus (a multiplicative R-valued cob ordism invariant characteristic of M ).

Every genus has a T k -equivariant extension
T k := · : U :T k - R [[u , . . . , u ]]. 1 k

We have
T k (M , c ) = (M ) + (g (M )) u . | |>0

In particular, the T k -equivariant extension of the universal genus ug = U U id : is ; hence the name "universal toric genus".
9


Rigidity and fibre multiplicativity. Consider fibre bundles M E âG M B , where M and B are connected and stably tangentially complex, G a compact Lie group of p ositive rank whose action preserves the stably complex structure on M , E B is a principal G-bundle. Then N := E âG M inherits a canonical stably complex structure. A genus : (N ) = (M if this holds
U R is multiplicative with resp ect to M whenever )(B ) for any such bundle ; for every M , then is fibre multiplicative.

k -rigid on M whenever T k : U :T k - R [[u , . . . , u ]] The genus is T 1 k k satisfies T (M , c ) = (M ); if this holds for every M , then is T k -rigid.

It follows that is T k -rigid whenever (G (M )) = 0 for | | > 0.
10


Other definitions of rigidity: Atiyah­Hirzebruch: Assume dex of an elliptic complex, k Then T is rigid if it takes tions).
k T can b e realised as the equivariant inT k : U :T k - RU (T k ). values in Z RU (T k ) (trivial representa-

Krichever: Considered Q-valued genera and equivariant extensions k Uk T : :T K (B T k ) Q. k Then T is rigid if it takes values in Q K (B T k ) Q. Our definition of rigidity extends b oth Atiyah­Hirzebruch's and Krichever's.

11


Thm 3. If the genus is T k -rigid on M , then it is mutiplicative with resp ect to M for bundles whose structure group G has the prop erty that U (B G) is torsion-free. On the other hand, if is multiplicative M , then it is T k -rigid on M . Pro of. Assume multiplicative. Apply to the bundle M G (M ) B . Since B b ounds for | | > 0, we have (G (M )) = 0, so is T k rigid. The other direction is proved by considering the pullback square E âG M - E G âG M E T k âT k M -

G

f

i



Tk



B

-

f

BG

-

i

BT k.

Ex 4. The signature is fibre multiplicative over any simply connected base, so it is a rigid genus.
12


Isolated fixed p oints. Assume Fix(M ) is isolated. Have T k -invariant c : M R2(l-n) . For any x Fix(M ), the sign (x) is +1 if the isomorphism
c,x p i x(M ) - x(M ) R2(l-n) - x = Cn Cl-n - Cn

resp ects the canonical orientations, and -1 if it do es not. If M is almost complex then (x) = 1 for every x Fix(M ). The non-trivial T k -representation Cn decomp oses into 1-dimensional representations as rx,1 . . . rx,n. wj (x) := (wj,1(x), . . . , wj,k (x)) the integral weight vector of rx,j . We refer to the collection of signs (x) and weight vectors wj (x) as the fixed p oint data for (M , c ).
13


Each weight vector determines a line bundle
wj (x) := wj,1(x) · · · wj,k (x) 1 k

over B T k , whose first Chern class is a formal p ower series [wj (x)](u) :=
2 in U (B T k ).




a [wj,1(x)](u1) 1 · · · [wj,k (x)](uk ) k
m cMU(j ) in 1 k ).

Here [m](uj ) denotes the p ower series 2 U (CP ), and the a are the co efficients of cMU(1 · · · 1 Mo dulo decomp osables we have that [wj (x)](u1, . . . , uk ) wj,1u1 + · · · + wj,k uk .

Thm 5 (Lo calisation formula). For any stably tangentially complex M 2n with isolated fixed p oints, the equation 1 (M ) = (x) j =1 [wj (x)](u) Fix(M )
- is satisfied in U 2n(B T k ).
14



n


Quasitoric manifolds revisited. Quasitoric complex T tions with explicit. manifolds M provide a vast source of examples of stably n -manifolds with isolated fixed p oints, for which calculathe fixed p oint data and Hirzebruch genera can b e made

Every such M is determined by the characteristic pair (P, ), where P is a simple n-p olytop e with m facets F1, . . . , Fm, is an integral n â m matrix. Given a fixed p oint x = Fj1 . . . Fjn denote N (P )x the matrix of column vectors normal to Fj1 , . . . , Fjn , x the square submatrix of of column vectors j1, . . . , jn, t Wx the matrix determined by Wx = -1 . x Prop 6. 1. the sign (x) is given by sign det(x N (P )x)
( )

2. the weight vectors w1(x), . . . wn(x) are the columns of Wx.
15


Elliptic genera. Buchstab er considered the formal group law d(u1) - d(u2) Fb(u1, u2) = u1c(u2) + u2c(u1) - au1u2 - u2u2 u1c(u2) - u2c(u1) 1 2 over the graded ring R = Z[a, cj , dk : j 2, k 1]/J , where deg a = 2, deg cj = 2j and deg dk = 2(k + 2); J is the ideal of asso ciativity relations, and c(u) := 1 +

j 2

cj uj ,

d(u) :=


k 1

dk uk .

Thm 7. The exp onential series fb(x) of Fb may b e written analytically as eax/(x, z ), where (x, z ) = (z - x) (z ) x e , (z ) (x)

(z ) is the Weierstrass sigma function, and (z ) = (ln (z )). Moreover, R Q is isomorphic to Q[a, c2, c3, c4] as graded algebras.
16


The function (x, z ) is known as the Baker­Akhiezer function asso ciated to the elliptic curve y 2 = 4x3 - g2x - g3. It satisfies the Lam´ equation, and is imp ortant in the theory of nonlinear integrable e equations.
U Krichever studied the genus kv : R corresp onding to the exp onential series fb, which therefore classifies the formal group law Fb. Analytically, it dep ends on the four complex variables z , a, g2 and g3. U Cor 8. Krichever's generalised elliptic genus kv : R induces an isomorphism of graded ab elian groups in dimensions < 10.

Thm 9. Let M 2n b e an S U -quasitoric manifold (i.e. c1(M ) = 0); then (1) the Krichever genus kv vanishes on M 2n, U (2) M 2n represents 0 in 2n whenever n < 5. Conjecture 10. Theorem 9(2) holds for all n.
17


Ex 11. 1. The 2-parameter To dd genus t2 may b e identified with the case c(u) = 1 - y z u2, d(u) = -y z (y + z )u - y 2z 2u2. The corresp onding formal group law is u + u2 - (y + z )u1u2 Ft2 = 1 . 1 - y z u1u2 It generalises the y -genus (z = -1) and the To dd genus (y = 0). 2. The elliptic genus E ll corresp onds to Euler's formal group law u c(u2) + u2c(u1) FE ll (u1, u2) = 1 1 - u2u2 12 u2 - u2 1 2 u2u2 , = u1c(u2) + u2c(u1) + u1c(u2) - u2c(u1) 1 2 and may therefore b e identified with the case a = 0, d(u) = -u2, and c2(u) = R(u) := 1 - 2 u2 + u4.
18


Further applications to rigidity. Prop 12. For any series f over a Q-algebra A, the corresp onding Hirzebruch genus f is T k -rigid on M only if the functional equation 1 (x) =c f (wj (x) · u) j =1 Fix(M ) is satisfied in A[[u1, . . . , uk ]], for some constant c A. The quasitoric examples CP 1, CP 2, and the T 2-manifold S 6 are all instructive. Ex 13. A genus f is T -rigid on CP 1 only if the equation 1 1 + = c, f (u) f (-u) holds in A[[u]]. The general analytic solution is u , where q (0) = 1. f (u) = 2 ) + cu/2 q (u An example is provided by the To dd genus, ftd(u) = (ez u - 1)/z . In fact td is multiplicative with resp ect to CP 1.
19



n


2 Ex 14. A genus f is T 2-rigid on the stably complex manifold CP(1,-1) only if the equation

1 1 1 - + =c f (u1)f (u2) f (u1)f (u1 + u2) f (-u2)f (u1 + u2) holds in A[[u1, u2]]. The general analytic solution satisfies f (u1) + f (u2) - cf (u1)f (u2) f (u1 + u2) = . 1 - cf (u1)f (u2) So f is the exp onential series of the 2-parameter To dd genus, with c = y + z and c = y z . Cor 15 (Musin). The 2-parameter To dd genus t2 is universal for rigid genera.

20


Ex 16. A genus f is T 2-rigid on the almost complex manifold S 6 only if the equation 1 1 + =c f (u1)f (u2)f (-u1 - u2) f (-u1)f (-u2)f (u1 + u2) holds in A[[u1, u2]], for some constant c. The general analytic solution is of the form eax/(x, z ), and f coincides with Krichever's exp onential series fb. Thm 17. Krichever's generalised elliptic genus kv is universal for genera that are rigid on S U -manifolds.

21


[1] Victor M. Buchstab er and Taras E. Panov. Torus Actions and Their Applications in Top ology and Combinatorics. Volume 24 of University Lecture Series, 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, 219­242; arXiv:math.AT/0609346.

[3] Victor M. Buchstab er, Taras E. Panov and Nigel Ray. Toric genera. Preprint (2009); arXiv:0908.3298.

22