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The Hirzebruch Genera of Complete Intersections 3 THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS JIANBO WANG, ZHIWANG YU, YUYU WANG Abstract. Following Brooks’s calculation of the Aˆ-genus of complete intersections, a new and more computable formula about the Aˆ-genus and α-invariant will be described as polynomials of multi-degree and dimension. We also give an iterated formula of Aˆ-genus and the necessary and sufficient conditions for the vanishing of Aˆ-genus of complex even dimensional spin complete intersections. Finally, we obtain a general formula about the Hirzebruch genus of complete intersections, and calculate some classical Hirzebruch genera as examples. 1. Introduction A genus of a multiplicative sequence is a ring homomorphism, from the ring of smooth compact manifolds (up to suitable cobordism) to another ring. A multiplicative se- quence is completely determined by its characteristic power series Q(x). Moreover, ev- ery power series Q(x) determines a multiplicative sequence. Given a power series Q(x), there is an associated Hirzebruch genus, which is also denoted as a ϕQ-genus. For more details, please see Section 2. The focus of this paper is mainly on the description of the Hirzebruch genus of complete intersections with multi-degree and dimension. n+r A complex n-dimensional complete intersection Xn(d1,...,dr) CP is a smooth complex n-dimensional manifold given by a transversal intersection⊂ of r nonsin- gular hypersurfaces in complex projective space. The unordered r-tuple d := (d1,...,dr) is called the multi-degree, which denotes the degrees of the r nonsingular hypersur- faces, and the product d := d1d2 dr is called the total degree. It is well-known arXiv:2003.02049v2 [math.AT] 1 May 2020 ··· that the diffeomorphism type of the real 2n-dimensional manifold Xn(d1,...,dr) de- pends only on the multi-degree and dimension. For the abbreviation of notation, set Xn(d) := Xn(d1,...,dr). By the Lefschetz hyperplane section theorem, the inclusion n+r 2 Xn(d) CP is n-connected. Let x H (Xn(d); Z) be the pullback of the first Chern class⊂ of the Hopf line bundle H (the∈ dual bundle of canonical line bundle) over Date: May 4, 2020. Key words and phrases. Hirzebruch genus, Aˆ-genus, alpha invariant, virtual Hirzebruch genus, com- plete intersection. This project was supported by the Natural Science Foundation of Tianjin City of China, No. 19JCYBJC30300. 1 2 JIANBOWANG,ZHIWANGYU,YUYUWANG 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 intersec- tion 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 < <k 6r X 1 1 X··· j 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 Combine the above theorem, we can give the α-invariant of complete intersections r Xn(d1,...,dr) with c1 = n + r +1 di even. − i=1 P Theorem 1.2. Let d = (d1,...,dr). Then for a complete intersection Xn(d) with r c1 = n + r +1 di even, the α-invariant α(Xn(d)) is equal to − i=1 P (1) Aˆ(Xn(d)), if n =0 (mod 4); 1 c 1+ d + + d (2) 2 1 − k1 ··· kj (mod 2), if n =1 (mod 4); n + r 06j6r−1 16k1<X···<kj 6r−1 1 ˆ (3) 2 A(Xn(d)), if n =2 (mod 4); (4) 0, if n =3 (mod 4). 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.
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