The Hirzebruch Genera of Complete Intersections 3

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n+r CP , the total Chern class and total Pontrjagin class of Xn(d) ([16]) are given by r c(X (d)) = (1+ x)n+r+1 (1 + d x)−1 , (1.1) n · i · i=1 Yr n+r+1 −1 p(X (d)) = 1+ x2 1+ d2 x2 . (1.2) n · i · i=1 Y In particular, the first Chern class is r c (X (d)) = n + r +1 d x. (1.3) 1 n − i · i=1 X n In addition, one can show that the evaluation of x on the fundamental class of Xn(d) is the total degree d = d d : 1 ··· r xn = d. ZXn(d) Note that, for a complete intersection Xn(d), the i-th Pontrjagin class pi must be an 2i 2 integral multiple of x , where x H (Xn(d); Z) = Z is a generator (n = 2). This ∈ ∼ 4i 6 multiple is independent of the choice of the generator x since pi H (Xn(d); Z), so we can compare the Pontrjagin classes of complete intersections with∈ the same dimension and different multi-degrees. Related to the diffeomorphism classification of complete intersections, there is the following conjecture, often called the Sullivan conjecture after Dennis Sullivan.

Sullivan conjecture. When n = 2, two diﬀerent complete intersections Xn(d) and ′ 6 Xn(d ) are diﬀeomorphic if and only if they have the same total degree, Pontrjagin classes and Euler characteristics. The partial known results on the Sullivan conjecture for complete intersections can be found in [6, 8, 13, 20]. The latest progress by Crowley and Nagy proves the Sullivan conjecture for n = 4 in [5].

Brooks [3] calculated the Aˆ-genus of complex hypersurfaces X2n(d1) and complete intersections X2n(d) with d =(d1,...,dr): n 2 d Aˆ(X (d )) = 1 j , (1.4) 2n 1 (2n + 1)! 2 − j=−n Y r 1 1 di− r di 1 n− 2 − P 2 ((1 + z) 1) ˆ i=1 i=1 A(X2n(d)) = (1 + z) 2n+r+1 − dz, (1.5) 2π√ 1 Γ · z − I (0) Q 1 where Γ (0) denotes a small circle around 0, and f(z)dz means the residue 2π√ 1 − IΓ (0) of f(z) at z = 0. Note that the Aˆ-genus of complex odd dimensional complete intersection is zero. (1.4) can also be found in [15, page 298]. Evidently, (1.4) is better relative to (1.5). One starting point for writing this paper is to give an explicit description of THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 3

Aˆ(X2n(d)), which might be simpler and more computable than (1.5). After all, (1.5) looks more cumbersome. The ﬁrst main theorem in this paper is as follows:

Theorem 1.1. The Aˆ-genus of complete intersection X2n(d)= X2n(d1,...,dr) is

r 1 c 1+ d + + d Aˆ(X (d)) = ( 1)r−j 2 1 − k1 ··· kj , 2n − 2n + r j=0 6k <

Note that for n> 1, the complete intersection Xn(d) is spin if and only if c1 is even. The complex one dimensional complete intersection X1(d) is a closed orientable surface with the second Stiefel-Whitney class ω2 = 0, and hence is spin. In [9], Hirzebruch deﬁned the virtual index and virtual generalized Todd genus for 2 a virtual submanifold (v1,...,vr) of M, where v1,...,vr H (M; Z), and M is a compact oriented manifold or a compact almost complex manifold.∈ Similarly, we deﬁne the virtual Hirzebruch genus of (v1,...,vr), which is equal to the Hirzebruch genus of a submanifold V with codimension 2r of M. As an application, an iterated formula about Aˆ-genus of complete intersections is given as follows:

Theorem 1.3. For any positive integers d1,...,dr, assume that dr−1 > dr > 2, then we have

dr−1 Aˆ(X (d ,...,d ,d )) = Aˆ(X (d ,...,d ,d + d 1 2k)). 2n 1 r−1 r 2n 1 r−2 r−1 r − − k X=0 4 JIANBOWANG,ZHIWANGYU,YUYUWANG

Brooks shows that, for a spin complete intersections X2n(d1,...,dr), its Aˆ-genus r vanishes if and only if c1 > 0 ([3]), where c1 = 2n + r + 1 di. We can use − i=1 Theorem 1.3 to give a new proof for the vanishing of Aˆ(X2n(d1,...,dPr)). As another application of virtual Hirzebruch genus, we give a general description of any Hirzebruch genus of complete intersections:

Theorem 1.4. For the given power series Q(x) = x/R(x), the Hirzebruch genus ϕQ on the complete intersection Xn(d) with d =(d1,...,dr) satisfies that: r 1 i=1 R(diz) ϕQ(Xn(d)) = n+r+1 dz. 2π√ 1 Γ (R(z)) − I (0) Q When this paper is near completion, we notice Baraglia’s paper [2]. Baraglia also give the similar description of the Aˆ-genus and alpha invariant of complete intersections by the different methods. Ours results in Theorem 1.1 and 1.2 are equivalent to the results in [2, Theorem 1.3 & 1.2]. This paper is organized as follows. In Section 2, we introduce some necessary concepts and properties on Hirzebruch genus. In Section 3, we prove Theorem 1.1 and Theo- rem 1.2. In Section 4, the virtual Hirzebruch genus is defined. As an application of virtual genus, we prove Theorem 1.3 and determine when the Aˆ-genus of complex even dimensional spin complete intersections vanishes. In Section 5, Theorem 1.4 is proved by using the virtual Hirzebruch genus of complete intersections, and we calculate some classical Hirzebruch genera of complete intersections as examples.

2. Hirzebruch genus For references about the Hirzebruch genus, [4, Appendix E], [9] and [10, 1.6-1.8] are recommended. Unless otherwise stated, all the manifolds mentioned in this§ paper are smooth, oriented, closed, and connected. A general real n-dimensional manifold is denoted by M n. We omit the real dimension n when there is no danger of ambiguity. If M admits an almost complex structure, it must be even dimensional. For an almost 2n complex manifold M , let T M be the complex tangent n-vector bundle, and (T M)R be the underlying real tangent 2n-vector bundle of M. For a diﬀerentiable manifold M n, T M is the tangent n-vector bundle of M. Every homomorphism ϕ : ΩU Λ from the complex bordism ring to a commutative ring Λ with unit can be regarded→ as a multiplicative characteristic of manifolds which is an invariant of bordism classes. Such a homomorphism is called a (complex) Λ-genus. The term multiplicative genus is also used, to emphasize that such a genus is a ring homomorphism. Assume that Λ does not have additive torsion, then every Λ-genus is fully determined by the corresponding homomorphism ΩU Q Λ Q, which is also denoted by ϕ. Every homomorphism ϕ : ΩU Q Λ⊗ Q→can⊗ be interpreted as a sequence of homogeneous polynomials K (c⊗,...,c→ ), i ⊗> 0 , deg K = 2i, with certain imposed { i 1 i } i THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 5 conditions. The conditions may be described as follows: an identity 1+ c + c + =(1+ c′ + c′ + ) (1 + c′′ + c′′ + ) 1 2 ··· 1 2 ··· · 1 2 ··· implies that K (c ,...,c )= K (c′ ,...,c′ ) K (c′′,...,c′′). (2.1) n 1 n i 1 i · j 1 j n>0 i>0 j>0 X X X A sequence of homogeneous polynomials = Ki(c1,...,ci), i > 0 with K0 = 1 is called multiplicative Hirzebruch sequenceK { if satisﬁes the (2.1} ). Write the following abbreviated notation: K K(c ,c ,... ) := 1 + K (c )+ K (c ,c )+ , 1 2 1 1 2 1 2 ··· then (2.1) is K(c ,c ,... )= K(c′ ,c′ ,... ) K(c′′,c′′,... ). 1 2 1 2 · 1 2 Moreover, a multiplicative sequence is one-to-one correspondence with a power series 2 K Q(x)=1+ q1x + q2x + Λ Q[[x]], where the notation Λ Q[[x]] means the ring of power series in x over··· ∈ the ring⊗ Λ Q, x = c , and q = K ⊗(1, 0,..., 0) (see [4, ⊗ 1 i i Appendix E] or [9, 1]), i.e. Q(x)=1+ K1(x, 0,..., 0) + K2(x, 0,..., 0)+ . By (2.1), an equality§ ··· 1+ c + + c = (1+ x ) (1 + x ) 1 ··· n 1 ··· n implies the equality Q(x ) Q(x )=1+ K (c )+ + K (c ,...,c )+ K (c ,...,c , 0) + 1 ··· n 1 1 ··· n 1 n n+1 1 n ··· = K(c1,...,cn).

It follows that the n-th term Kn(c1,...,cn) in the multiplicative Hirzebruch sequence U = Ki(c1,...,ci) corresponding to a genus ϕ : Ω Q Λ Q is the degree 2n part K { n } ⊗ → ⊗ of the series i=1 Q(xi) Λ Q[[c1,...,cn]]. Follow the statement in [4, Appendix E], n ∈ ⊗ the series i=1 Q(xi) can be regarded as a universal characteristic class of a complex n-vector bundles.Q For example, for a complex n-vector bundle ξ with the total Chern class c(ξ)=1+Q c + + c = (1+ x ) (1 + x ), where x ,...,x are the formal 1 ··· n 1 ··· n 1 n roots of c(ξ), we can deﬁne the ϕQ-class of ξ by n

ϕQ(ξ)= K(c1,...,cn)= Q(xi). (2.2) i=1 Y Deﬁnition 2.1. Let Q(x) be a power series with constant term being one. For an 2n almost complex manifold M , let x1,...,xn be the formal roots of the total Chern class of T M, i.e., c(T M)=(1+ x ) (1+ x ). The Hirzebruch genus ϕ corresponding 1 ··· n Q to Q(x) is deﬁned by n

ϕQ(M)= ϕQ(T M)= Q(xi), M M i=1 Z Z Y 6 JIANBOWANG,ZHIWANGYU,YUYUWANG where M ϕQ(T M)= ϕQ(T M), [M] is the evaluation of the degree 2n part of ϕQ(T M) on the fundamental class [M] of M. R A parallel theory of genera exists for oriented manifolds. These genera are homo- morphisms ΩSO Λ from the oriented bordism ring to Λ. Once again we assume that the ring Λ→ does not have additive torsion. Then every oriented Λ-genus ϕ is determined by the corresponding homomorphism ΩSO Q Λ Q, which we also denote by ϕ. Such genera are in one-to-one correspondence⊗ → with⊗ even power series 2 4 Q(x)=1+ q2x + q4x + Λ Q[[x]]. Similarly as the complex···∈ case⊗ above, for a real n-vector bundle ξ with the total 2 2 2 Pontrjagin class p(ξ)=1+ p1 + + p n = (1+ x1)(1+ x2) (1+ x n ), we can deﬁne ··· [ 4 ] ··· [ 4 ] the ϕQ-class of ξ by

n [ 4 ]

ϕQ(ξ)= K(p1,...,p n )= Q(xi). (2.3) [ 4 ] i=1 Y Definition 2.2. For a compact, oriented, differentiable manifold M of dimension n, the Hirzebruch genus ϕQ corresponding to an even power series Q(x) is defined by

n [ 4 ]

ϕQ(M)= ϕQ(T M)= Q(xi), M M i=1 Z Z Y 2 2 where the x1,...,x n are the formal roots of the total Pontrjagin class of T M, i.e., [ 4 ] 2 2 p(T M)=(1+ x1) (1 + x n ). ··· [ 4 ] An oriented genus ϕ : ΩSO Λ deﬁnes a complex genus given by the composition ϕ → ΩU ΩSO Λ with the homomorphism ΩU ΩSO forgetting the stably complex structure.→ Since−→ ΩU ΩSO is onto modulo torsion,→ and Λ is assumed to be torsion-free, one loses no information→ by passing from an oriented genus to a complex one. On the other hand, a complex genus ϕ : ΩU Λ factors through an oriented genus ΩSO Λ whenever its corresponding power series→ Q(x) is even. →

3. The Aˆ-genus and α-invariant of complete intersections For a compact oriented diﬀerentiable manifold M 4k, Hirzebruch ([9, Page 197]) de- ﬁned the Aˆ-sequence of M 4k as a certain polynomials in the Pontrjagin classes of M 4k. More concretely, the power series

1 2 4 6 8 2 x x 7x 31x 127x Q(x)= 1 =1 + + + sinh 2 x − 24 5760 − 967680 154828800 ··· THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 7 deﬁnes a multiplicative sequence Aˆj(p1,...,pj) . For example, the ﬁrst four terms of the sequence are as follows (for abbreviation,n pj o:= pj(M)= pj(T M)): p Aˆ = 1 , 1 −24 7p2 4p Aˆ = 1 − 2 , 2 5760 31p3 + 44p p 16p Aˆ = − 1 1 2 − 3 , 3 967680 381p3 904p2p + 208p2 + 512p p 192p Aˆ = 1 − 1 2 2 1 3 − 4 . 4 464486400 ˆ ˆ The A-genus A(M) is the ϕQ-genus with the power series Q(x) as above, and

Aˆ(M)= Aˆ(p1,...,pk), ZM ˆ ˆ where A(p1,...,pk)= ϕQ(T M) is the A-class deﬁned as in Section 2. It is a result of Lichnerowicz that the Aˆ-genus must vanish for a compact spin manifold which admits a Riemannian metric with positive scalar curvature (see [17]). In [11], Hitchin generalized this result to α(M) = 0 if M has a metric of positive scalar Spin −∗ curvature, where α : Ω∗ KO (pt) is a homomorphism, and the α-invariant α(M) KO−n(pt) is a KO-characteristic→ number for a spin manifold M of dimension ∈ n. This is a strict generalization of the result of Lichnerowicz since α(M) = Aˆ(M) 1 ˆ if n = 0 (mod 8) and α(M) = 2 A(M) if n = 4 (mod 8). Furthermore, Stolz showed that a spin manifold M of dimension n > 5 admits a positive scalar curvature metric if and only if α(M) vanishes (see [18]), which proved the Gromov-Lawson conjecture. By virtue of the α-invariants of complex 4k +1-dimensional complete intersection X4k+1(d) (k > 1) and Seiberg-Witten theory in complex dimension 2, Fang and Shao in [7] gave a classiﬁcation list of complete intersections admitting Riemannian metrics with positive scalar curvature. The Aˆ-genus is a special case of elliptic genera. There are many results concerning the Aˆ-genus which are related with group actions (see [12]). From the Atiyah-Singer index theorem, the Aˆ(M) can be interpreted as the index of the Dirac operator acting on spin-bundles over M. A profound development of the classical result by Atiyah- Hirzebruch (see [1]): If M is connected spin and admits a nontrivial smooth S1-action, then the Aˆ-genus Aˆ(M) vanishes. Based on the Brooks’s calculation (1.5), we can obtain a more computable formula about the Aˆ-genus of complex even dimensional complete intersection.

Theorem 3.1. The Aˆ-genus of complete intersection X2n(d)= X2n(d1,...,dr) is r 1 c 1+ d + + d Aˆ(X (d)) = ( 1)r−j 2 1 − k1 ··· kj , 2n − 2n + r j=0 6k <

r where c1 =2n + r +1 di, i.e. c1 x is the ﬁrst Chern class c1(X2n(d)) as in (1.3), − i=1 · a a(a 1)(a 2) P (a k + 1) := − − ··· − , and d + + d vanishes if j =0. k k! k1 ··· kj Proof. Let Γ (0) denote a small circle around 0. By (1.5)

r di r 1 ((1 + z) 1) n 1 di− 1 − 2 − P 2 i=1 − ˆ i=1 A(X2n(d)) = (1 + z) 2n+r+1 dz 2π√ 1 Γ (0) · Q z − I r r d 1 n− 1 − P i− +d +···+d 1 1 r j 2 2 k1 kj = ( 1) − (1 + z) i=1 dz √ z2n+r+1 · − 2π 1 Γ (0) j=0 6k <

2n+r Thus, by the residue theorem, the Aˆ-genus of X2n(d) is the coeﬃcient of z in the r 1 c1−1+d +···+d double summation ( 1)r−j (1 + z) 2 k1 kj . We have j=0 − 16k1<···

r 1 c 1+ d + + d Aˆ(X (d)) = ( 1)r−j 2 1 − k1 ··· kj . 2n − 2n + r j=0 6k <

Thus Theorem 3.1 is done.

We can give another proof of Theorem 3.1 by calculating the Aˆ-genus directly.

2nd Proof of Theorem 3.1. Firstly, for any almost complex manifold M, the Aˆ-class of the underlying real tangent bundle (T M)R is

∞

Aˆ((T M)R)= Aˆk(p1,...,pk). (3.1) k X=0 THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 9

By (1.2) and (2.3), the Aˆ-class of (T (Xn(d)))R is

n+r+1 r di 1 sinh 2 x 2 x Aˆ((T (Xn(d)))R)= di x · sinh 1 x i=1 2 2 ! Y n+r+1 r di di exp 2 x exp 2 x x = − − 1 1 dix · exp x exp x i=1 2 2 ! Y − − r di di 1 n+1 i=1 exp 2 x exp 2 x = x − −n r d · exp 1 x exp 1 x + +1 Q 2 − − 2 r r 1 n+1 1 1 i=1 (exp (dix) 1) = x exp (n + r + 1)x dix − d · 2 − 2 · (exp (x) 1)n+r+1 i=1 ! Q r X − 1 n+1 1 i=1 (exp (dix) 1) = x exp c1x − , (3.2) d · 2 · (exp (x) 1)n+r+1 Q − r where d = d1 dr is the total degree of Xn(d) and c1 = n + r +1 di. ··· − i=1 By (3.2), we have P

r 1 2n+1 1 i=1 (exp (dix) 1) Aˆ((T (X2n(d)))R)= x exp c1x − . d · 2 · (exp (x) 1)2n+r+1 Q − Then

Aˆ(X2n(d)) = Aˆ((T (X2n(d)))R) ZX2n(d) 1 r 1 exp c1x = x2n+1 2 (exp(d x) 1). d · (exp(x) 1)2n+r+1 i − X2n(d) i=1 Z − Y Since what we are interested in is some coeﬃcient in a polynomial, we use residue theorem to calculate this integral. Notice that x2n = d is the total degree of the X2n(d) complete intersection X (d), then we have 2n R

exp 1 c z r ˆ 1 2 1 A(X2n(d)) = 2n+r+1 (exp(diz) 1)dz 2π√ 1 Γ (exp(z) 1) − − I (0) − i=1 1 Yr 1 exp ( 2 c1 1)z = − (exp(diz) 1)d exp(z). 2π√ 1 (exp(z) 1)2n+r+1 − Γ (0) i=1 − I − Y 10 JIANBOWANG,ZHIWANGYU,YUYUWANG

By means of variable substitution: ω = exp(z) 1, − 1 r 2 c1−1 1 (1 + ω) di Aˆ(X2n(d)) = ((1 + ω) 1)dω 2π√ 1 ω2n+r+1 − − IΓ (0) i=1 r Y 1 c 1+ d + + d = ( 1)r−j 2 1 − k1 ··· kj . − 2n + r j=0 6k <

Proof. (1) follows directly from [11, Proposition 4.2], and Aˆ(Xn(d)) is as in Theorem 3.1. (3) is trivial. For (2), when n =1 (mod 4), by [7, (8)], the α-invariant of Xn(d) is as follows: 1 (n + r 1 d d + d ) α(X (d)) = 2 − ± 1 ±···± r−1 r (mod 2), (3.3) n n + r X where the summation sums over all the possibilities d d +d . By [7, Remark ± 1 ±···± r−1 r 2], the choice of dr in (3.3) is not important to the result, since n−1 + k n−1 k 2 = 2 − (mod 2). n n THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 11

Therefore we can make (3.3) stepped forward: 1 (n + r 1 d d + d ) α(X (d)) = 2 − ± 1 ±···± r−1 r (mod 2) n n + r X 1 (n + r +1 d d d ) 1 = 2 ± 1 ±···± r−1 − r − (mod 2) n + r X r 1 2 (n + r +1 di + d1 d1 + d2 d2 + + dr−1 dr−1) 1 = − i=1 ± ± ··· ± − P n + r X 1 c + 1 (d d + d d + + d d ) 1 = 2 1 2 1 ± 1 2 ± 2 ··· r−1 ± r−1 − n + r X 1 c 1+ ε d + ε d + + ε d = 2 1 − 1 1 2 2 ··· r−1 r−1 n + r ε1,...,εXr−1∈{0,1} r −1 1 c 1+ ε d + ε d + + ε d = 2 1 − 1 1 2 2 ··· r−1 r−1 n + r j=0 j=ε1+···+εr−1 X ε1,...,εXr−1∈{0,1} r −1 1 c 1+ d + + d = 2 1 − k1 ··· kj . n + r j=0 6k <

α(Xn(d1,...,dr−1,dr))

dr = Aˆ((T (X (d ,...,d )))R) exp x (mod 2). n+1 1 r−1 · 2 ZXn+1(d1,...,dr−1) Begin with this formula, we can directly prove Theorem 3.2 (2).

2nd Proof of Theorem 3.2 (2).

α(Xn(d1,...,dr−1,dr))

dr = Aˆ((T (X (d ,...,d )))R) exp x (mod 2) n+1 1 r−1 · 2 ZXn+1(d1,...,dr−1) dr = Aˆ((T (X (d ,...,d )))R) exp x (mod 2). n+1 1 r−1 · − 2 ZXn+1(d1,...,dr−1) 12 JIANBOWANG,ZHIWANGYU,YUYUWANG

By (3.2), we have

α(Xn(d1,...,dr−1,dr)) xn+2 r−1 exp di x exp di x d = i=1 2 − − 2 exp r x . r−1 1 1 n+r+1 Xn+1(d1,...,dr−1) di · exp x exp x · − 2 Z i=1 Q 2 − − 2 By residue theorem, Q

α(Xn(d1,...,dr−1,dr)) 1 r−1 exp di z exp di z d = i=1 2 − − 2 exp r z dz 1 1 n+r+1 2π√ 1 Γ (0) exp z exp z · − 2 − I Q 2 − − 2 r r−1 1 1 di i=1 (exp (diz) 1) = exp (n + r + 1)z exp z − dz √ 2 · − 2 · (exp (z) 1)n+r+1 2π 1 Γ (0) i=1 − I Y Q − r r−1 1 1 i=1 (exp (diz) 1) = exp (n + r +1 di)z n+−r+1 dz 2π√ 1 Γ 2 − · (exp (z) 1) − I (0) i=1 ! Q − r−1 X 1 1 i=1 (exp (diz) 1) = exp c1z n+−r+1 exp( z)d(exp(z) 1). 2π√ 1 Γ 2 · (exp (z) 1) · − − − I (0) Q − By means of variable substitution: ω = exp(z) 1, − α(Xn(d1,...,dr−1,dr))

r−1 di 1 1 c ((1 + ω) 1) = (1 + ω) 2 1 i=1 − (1 + ω)−1dω 2π√ 1 · ωn+r+1 · − IΓ (0) Q r−1 di 1 1 c ((1 + ω) 1) = (1 + ω) 2 1−1 i=1 − dω 2π√ 1 · ωn+r+1 − IΓ (0) Q 1 c 1+ d + + d = 2 1 − k1 ··· kj (mod 2). n + r 06j6r−1 16k1

4. The virtual Hirzebruch genus of complete intersections Inspired by the virtual index and virtual generalized Todd genus in [9, 9, 11], we define the virtual Hirzebruch genus. First of all, let’s introduce the concept§ of§ virtual submanifold. The following part about the virtual submanifold is in [10, 3.1] According to Thom [19], every 2-dimensional integral cohomology class v§ H2(M n; Z) of a compact oriented differentiable manifold M n can be represented by a∈ submanifold V n−2, i.e., the cohomology class v is the Poincaréduality of the fundamental class of the oriented submanifold V n−2. Therefore v H2(M; Z) is called a virtual submanifold. 2 ∈ Assume that v1, v2,...,vr H (M; Z), and v1 is a virtual submanifold represented by n−2 ∈ n−2 a submanifold V of the manifold M, that the restriction of v2 to V represents n−4 n−2 n−2(r−1) a submanifold V of V , ... , and finally that the restriction of vr to V THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 13 represents a submanifold V n−2r of V n−2(r−1). The manifold V n−2r is then a submanifold of M of codimension 2r, which represents the virtual submanifold (v1,...,vr). An alternative construction is the following: If v1,...,vr can be represented by codimen- ′ ′ sion 2 submanifolds V1 ,...,Vr of M, which are transversal to one another, then their transversal intersection represents the virtual submanifold (v1,...,vr). Let’s set up the following convention: (1) If M is a compact almost complex manifold, Q(x) = x/R(x) is taken as any power series with constant term being one. (2) If M is a compact oriented differentiable manifold, Q(x) = x/R(x) is taken as an even power series with constant term being one. 2 Definition 4.1. For the integral cohomology classes v1, v2,...,vr of H (M; Z), the virtual Hirzebruch genus ϕQ of the virtual submanifold (v1, v2,...,vr) is defined as follows: r ϕ (v ,...,v ) = R(v ) ϕ (T M), Q 1 r M j · Q M j=1 Z Y where ϕQ(T M) is the ϕQ-class defined as in Section 2.

In fact, the virtual Hirzebruch genus of the virtual submanifold (v1, v2,...,vr) is related to the Hirzebruch genus of the submanifold V n−2r of M n. Similar to the argument in [9, 9, 11], we have the following result. § § Proposition 4.2. As the notations in Deﬁnition 4.1. Assume that the virtual subman- n−2r n ifold (v1,...,vr) is represented by a submanifold V of the manifold M , then n−2r ϕQ(v1,...,vr)M = ϕQ(V ). Proof. We only give the proof when M is a compact oriented diﬀerentiable n-manifold. This proof is adapted to the case of almost complex manifold (see [9, 11]). Let v H2(M n; Z) be a virtual manifold represented by a submanifold§ V n−2, and j : V n−2 ∈ M n be the embedding of the oriented submanifold V n−2 of M n. Let TV n−2 and T M n→be tangle bundles of V n−2, M n respectively and is the normal bundle of V n−2 in M n, then N j∗(T M n)= TV n−2 . ⊕N Let p(V n−2) and p(M n) be the total Pontrjagin classes of V n−2 and M n respectively. Then j∗p(M n)= p(V n−2)p( ) modulo torsion. N Since p( )= j∗(1 + v2), then N p(V n−2)= j∗((1 + v2)−1p(M n)). Note that (1+v2)−1 is unique in the cohomology class of M n. Thus, for every Hirzebruch genus ϕQ associated to an even power series Q(x), we have R(v) ϕ (TV n−2)= j∗ ϕ (T M n) . Q v Q 14 JIANBOWANG,ZHIWANGYU,YUYUWANG

For a cohomology class u Hn−2(M n; Z), by Poincar´eduality, we have ∈ j∗(u)= vu. n−2 n ZV ZM Therefore,

n−2 n−2 ∗ R(v) n ϕQ(V )= ϕQ(TV )= j ϕQ(T M ) n−2 n−2 v ZV ZV n = R(v)ϕQ(T M )= ϕQ(v)M . n ZM n−2r By ﬁnite induction, it implies that ϕQ(v1,...,vr)M = ϕQ(V ).

n+r Now we consider the complete intersection Xn(d1,...,dr) embedded in CP . Let x H2(CP n+r; Z) be the ﬁrst Chern class of the Hopf line bundle H over CP n+r, ∈ then the virtual submanifold (d1x,...,drx) is represented by the complete intersection Xn(d1,...,dr). So by Proposition 4.2, we have

Corollary 4.3. The ϕQ-genus of Xn(d1,...,dr) is the virtual ϕQ-genus of the virtual submanifold (d1x,...,drx), i.e.,

ϕQ(d1x,...,drx)CP n+r = ϕQ(Xn(d1,...,dr)).

As a special case of Deﬁnition 4.1, we can also deﬁne the virtual Aˆ-genus.

Definition 4.4. Let M be a compact oriented differentiable manifold. For v1,...,vr 2 ∈ H (M; Z), the virtual Aˆ-genus of (v1,...,vr) is defined as follows: r Aˆ(v ,...,v ) = R(v ) Aˆ(T M), 1 r M j · M j=1 Z Y 1 x 2 x where Q(x)= = 1 . R(x) sinh 2 x Proposition 4.5. Let M be a compact oriented differentiable manifold. For any v ,...,v ,w H2(M; Z), the virtual Aˆ-genus satisfies the following equality: 1 r ∈

Aˆ(v1,...,vr−2, vr−1 + w, vr + w)M

= Aˆ(v1,...,vr−2, vr−1, vr)M + Aˆ(v1,...,vr−2, vr−1 + vr + w,w)M . 1 x ˆ x 2 Proof. The A-genus is corresponding to the power series Q(x) = = 1 , R(x) sinh 2 x and R(x) = 2 sinh 1 x satisﬁes the following identity 2 R(u +w)R(v + w)= R(u)R(v)+ R(u + v + w)R(w). THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 15

By Deﬁnition 4.4, for any w H2(M; Z), it induces the following equality ∈ Aˆ(v1,...,vr−2, vr−1 + w, vr + w)M r−2 = R(v ) R(v + w) R(v + w)Aˆ(T M) j · r−1 · r M j=1 Z Y r r−2 = R(v ) Aˆ(T M)+ R(v ) R(v + v + w) R(w)Aˆ(T M) j · j · r−1 r · M j=1 M j=1 Z Y Z Y = Aˆ(v1,...,vr−2, vr−1, vr)M + Aˆ(v1,...,vr−2, vr−1 + vr + w,w)M . By Corollary 4.3 and Proposition 4.5, we have

Corollary 4.6. For any positive integers d1,...,dr, e, we have the following equality on the Aˆ-genus of complete intersections

Aˆ(X2n(d1,...,dr−2,dr−1 + e, dr + e))

= Aˆ(X2n(d1,...,dr−1,dr)) + Aˆ(X2n(d1,...,dr−2,dr−1 + dr + e, e)).

Theorem 4.7. For any positive integers d1,...,dr, assume that dr−1 > dr > 2, then we have

dr−1 Aˆ(X (d ,...,d ,d )) = Aˆ(X (d ,...,d ,d + d 1 2k)). (4.1) 2n 1 r−1 r 2n 1 r−2 r−1 r − − k X=0 Note that, if dr =1, Aˆ(X2n(d1,...,dr−1, 1)) = Aˆ(X2n(d1,...,dr−1)). Proof. Applying Corollary 4.6, then

Aˆ(X2n(d1,...,dr−2,dr−1,dr)) = Aˆ(X (d ,...,d ,d 1,d 1)) + Aˆ(X (d ,...,d ,d + d 1, 1)) 2n 1 r−2 r−1 − r − 2n 1 r−2 r−1 r − = Aˆ(X (d ,...,d ,d 1,d 1)) + Aˆ(X (d ,...,d ,d + d 1)). 2n 1 r−2 r−1 − r − 2n 1 r−2 r−1 r − Let’s iterate the above process, the proof is ﬁnished.

Consider complex even dimensional spin complete intersection X2n(d) with multi- r degree d =(d1,...,dr). Note that X2n(d) is spin if and only if c1 =2n+r+1 di =0 −i=1 (mod 2). In [3, Theorem 2], Brooks can determine when Aˆ(X2n(d)) = 0 forP the spin complete intersections X2n(d). Now, we give a new proof for the necessary and suﬃcient conditions related to the vanishing of Aˆ(X2n(d)).

Theorem 4.8. For each complex even dimensional spin complete intersection X2n(d) with d =(d1,...,dr), the Aˆ-genus of X2n(d) has the following properties:

(1) Aˆ(X2n(d))=0, if c1 > 0; (2) Aˆ(X2n(d)) > 0, if c1 6 0, 16 JIANBOWANG,ZHIWANGYU,YUYUWANG

r where c1 =2n + r +1 di. − i=1 P Proof. Since the diﬀeomorphism type of X2n(d) is independent of the order of the degrees d1,...,dr−1,dr, without losing generality, we assume that dr−1 > dr. We prove the theorem by induction on r. n ˆ 2 d1 For r = 1, A(X2n(d1)) = (2n+1)! 2 j , c1 =2n+2 d1, and this case is trivial. j=−n − − For r> 1, if c1 > 0, then Q

r r−2 2n + r +1 d =2n +(r 1)+1 d +(d + d 1) > 0, − i − − i r−1 r − i=1 i=1 ! X r−2 X 2n +(r 1)+1 d +(d + d 1 2k) > 0, 0 6 k 6 d 1. − − i r−1 r − − r − i=1 ! X Thus, by induction hypothesis,

Aˆ(X (d ,...,d ,d + d 1 2k))=0, 0 6 k 6 d 1, 2n 1 r−2 r−1 r − − r − i.e. the right side of (4.1) is zero, then Aˆ(X2n(d1,...,dr)) = 0; Conversely, if c1 6 0, then

r r−2 2n + r +1 d =2n +(r 1)+1 d +(d + d 1) 6 0. − i − − i r−1 r − i=1 i=1 ! X X Thus, by induction hypothesis,

Aˆ(X (d ,...,d ,d + d 1)) > 0, 2n 1 r−2 r−1 r − and all other summation terms in (4.1) are nonnegative, so Aˆ(X2n(d1,...,dr)) > 0.

5. Calculation of classical Hirzebruch genera of complete intersections In this section, for the power series Q(x)= x/R(x), we discuss the associated Hirze- bruch genus ϕQ of complete intersections Xn(d) with multi-degree d = (d1,...,dr). Then for certain given power series Q(x), we calculate the associated classical Hirze- bruch genera of complete intersections as examples.

Theorem 5.1. The Hirzebruch genus ϕQ on the complete intersection Xn(d) satisﬁes that: r 1 i=1 R(diz) ϕQ(Xn(d)) = n+r+1 dz. 2π√ 1 Γ (R(z)) − I (0) Q THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 17

Proof. Let x H2(CP n+r; Z) be the ﬁrst Chern class of the Hopf line bundle H over CP n+r. By Corollary∈ 4.3 and Deﬁnition 4.1,

ϕQ(Xn(d)) = ϕQ(d1x,...,drx)CP n+r r n+r = R(dix) ϕQ(T (CP )). n+r · CP i=1 Z Y The total Chern class of CP n+r is c(CP n+r)=(1+ x)n+r+1, so

n+r n+r+1 ϕQ(T (CP ))=(Q(x)) .

Thus

r n+r+1 ϕQ(Xn(d)) = R(dix) (Q(x)) n+r · ZCP i=1 Y r n+r+1 i=1 R(dix) = x n+r+1 . CP n+r (R(x)) Z Q

By residue theorem,

1 r R(d z) i=1 i ϕQ(Xn(d)) = n+r+1 dz. 2π√ 1 Γ (R(z)) − I (0) Q

Example 5.2. For Ty-genus (or generalized Todd genus) in [9, page 93], the associated x power series is Q(y; x)= , where R(y; x)

exp(x(y + 1)) 1 R(y; x)= − . exp(x(y +1))+ y

Hence by Theorem 5.1 we have

r 1 j=1 R(y; djz) Ty(Xn(d)) = n+r+1 dz 2π√ 1 Γ (R(y; z)) I (0) Q r − 1 exp(d z(y + 1)) 1 exp(z(y +1))+ y n+r+1 = j − dz. (5.1) √ exp(djz(y +1))+ y · exp(z(y + 1)) 1 2π 1 Γ (0) j=1 − I Y − 18 JIANBOWANG,ZHIWANGYU,YUYUWANG

Example 5.3. In Example 5.2, when y = 0, R(y; x) is R(0; x)=1 exp( x). The − − T0-genus is the Todd genus, which has the following form:

r 1 (1 exp ( djz)) j=1 − − Td(Xn(d)) = n+r+1 dz 2π√ 1 Γ (0) Q(1 exp ( z)) − I r − − 1 (1 exp ( djz)) j=1 − − = n+r+1 exp(z)d(1 exp ( z)) 2π√ 1 Γ (1 exp ( z)) · − − I (0) Q − r − − 1 1 (1 ω)dj = j=1 − − (1 ω)−1dω 2π√ 1 ωn+r+1 · − IΓ (0) Q − r 1 1 j −1+d +···+d = ( 1) (1 ω) k1 kj dω √ ωn+r+1 − − 2π 1 Γ (0) j=0 6k <

r 1 (1 exp ( djz)) j=1 − − Td(Xn(d)) = n+r+1 dz 2π√ 1 Γ (0) Q(1 exp ( z)) − I − −r 1 (exp (djz) 1) j=1 − = exp (c1z) n+r+1 dz 2π√ 1 Γ (0) Q(exp (z) 1) − I r − 1 j=1(exp (djz) 1) = exp (c1z) n+−r+1 exp( z)d(exp (z) 1) 2π√ 1 Γ (exp (z) 1) · − − I (0) Q − r − 1 (1 + ω)dj 1 = j=1 − (1 + ω)c1−1dω 2π√ 1 ωn+r+1 · IΓ (0) Q r − c 1+ d + + d = ( 1)r−j 1 − k1 ··· kj , (5.3) − n + r j=0 6k <

r dj z 1 j=1 1+dj z χ(Xn(d)) = dz. z n+r+1 2π√ 1 Γ (0) − I Q1+z z ω Let ω = , then z = . It implies that 1+ z 1 ω − d z d ω j = j , 1+ d z 1+(d 1)ω j j − dω = (1 ω)2dz (the diﬀerential of ω). − 20 JIANBOWANG,ZHIWANGYU,YUYUWANG

Hence, r 1 djω 1 1 χ(Xn(d)) = n+r+1 2 dω 2π√ 1 Γ (0) 1+(dj 1)ω · ω · (1 ω) − I j=1 − − r Y 1 1 1 = dj r 2 n+1 dω · 2π√ 1 Γ (0) (1+(d 1)ω) (1 ω) · ω j=1 − I j=1 j − · − Yr Q = d h (1, 1, 1 d ,..., 1 d ), j · n − 1 − r j=1 Y k n 1 where hn(a1,...,ak) = ai1 ai2 ain is the coeﬃcient of z in , 6 6 6 6 ··· 1 a z 1 i1 ··· in k j=1 − j and hn(a1,...,ak) is called aP complete homogeneous symmetric polynomialQ of degree n on indeterminates a1,...,ak. Example 5.5. In Example 5.2, when y = 1, R(y; x) is R(1; x) = tanh x and (5.1) is the corresponding signature of complete intersection Xn(d).

τ(Xn(d)) r 1 j=1 tanh(djz) = n+r+1 dz 2π√ 1 Γ (tanh z) I (0) Q − r 1 exp (2d z) 1 exp (2z)+1 n+r+1 = j − dz 2π√ 1 Γ exp (2djz)+1 · exp (2z) 1 − I (0) j=1 − Yr 1 exp (2d z) 1 exp (2z)+1 n+r+1 exp( 2z) = j − − d(exp(2z) 1) √ exp (2djz)+1 · exp (2z) 1 · 2 − 2π 1 Γ (0) j=1 − I Y − r n+r+1 1 (1 + ω)dj 1 2+ ω 1 = − dω. √ (1 + ω)dj +1 · ω · 2(1 + ω) 2π 1 Γ (0) j=1 − I Y Example 5.6. For Witten genus [10, page 82], the power series is that x x Q(x)= = , R(L; x) σL(x) where R(L; x) = σL(x) is the Weierstrass σ-function for Lattice L. Hence, by Theo- rem 5.1, the Witten genus of Xn(d) is r 1 j=1 σL(djz) W (Xn(d)) = n+r+1 dz. 2π√ 1 Γ (σ (z)) − I (0) Q L Example 5.7. For Aˆ-genus, the power series corresponding to Aˆ-genus is x 1 x 1 Q(x)= = 2 , where R(x) = 2 sinh x . R(x) sinh 1 x 2 2 THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS 21

Then by Theorem 5.1,

Aˆ(X2n(d1,...,dr)) r 1 1 2 sinh djz = j=1 2 dz 1 2n+r+1 2π√ 1 Γ (0) 2 sinh z − I Q 2 r 1 1 1 exp djz exp djz = j=1 2 − − 2 dz 1 1 2n+r+1 2π√ 1 Γ (0) exp z exp z − I Q 2 − − 2 r r 1 1 (exp (djz) 1) j=1 − = exp 2n + r +1 di z 2n+r+1 dz 2π√ 1 Γ (0) 2 − ! · (exp (z) 1) − I i=1 Q − r X 1 1 j=1(exp (djz) 1) = exp c1z 2n+−r+1 exp ( z)d(exp (z) 1) 2π√ 1 Γ (0) 2 · (exp (z) 1) · − − − I Q − r dj 1 1 c j=1 (1 + ω) 1 = (1 + ω) 2 1−1 − dω 2π√ 1 · ω2n+r+1 IΓ (0) Q r − 1 c 1+ d + + d = ( 1)r−j 2 1 − k1 ··· kj , − 2n + r j=0 6k <

r exp (kdiz)−1 1 i=1 k exp(diz) Ak(Xn(d)) = n+r+1 dz 2π√ 1 Γ (0) Qexp (kz)−1 − I k exp(z) r n+r+1 1 1 exp (d z) 1 k exp 1 z = i − k dz k · 2π√ 1 k exp 1 d z · exp (z) 1 Γ (0) i=1 k i ! − I Y − r n+r+1 1 1 exp (dz) 1 k exp 1 z = i − k exp( z)d(exp (z) 1) k · 2π√ 1 k exp 1 d z · exp (z) 1 − − Γ (0) i=1 k i ! − I Y − 22 JIANBOWANG,ZHIWANGYU,YUYUWANG

n+r+1 r d 1 1 1 (1 + ω) i 1 k(1 + ω) k = − (1 + ω)−1dω 1 d k · √ k i · ω · 2π 1 Γ (0) i=1 k(1 + ω) ! − I Y r di 1 ((1 + ω) 1) 1 n+r+1−P d −1 = kn i=1 − (1 + ω) k ( i i) dω · 2π√ 1 ωn+r+1 · Γ (0) Q − I r di 1 1 ((1 + ω) 1) c1−1 = kn i=1 − (1 + ω) k dω · 2π√ 1 ωn+r+1 · IΓ (0) Q r − 1 c 1+ d + + d = kn ( 1)r−j k 1 − k1 ··· kj , − n + r j=0 6k <

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(Jianbo Wang) School of Mathematics, Tianjin University, Tianjin 300350, China E-mail address: [email protected]

(Zhiwang Yu) School of Mathematics, Tianjin University, Tianjin 300350, China E-mail address: [email protected]

(Yuyu Wang) College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China E-mail address: [email protected]