Remarks on 5-dimensional complete intersections

This paper will give some examples of diffeomorphic complex 5-dimensional complete intersections and remarks on these examples. Then a result on the existence of diffeomorphic complete intersections that belong to components of the moduli space of different dimensions will be given as a supplement to the results of P.Br\"uckmann (J. reine angew. Math. 476 (1996), 209-215; 525 (2000), 213-217).


Introduction
Let X n (d) ⊂ CP n+r be a smooth complete intersection of multidegree d := (d 1 , · · · , d r ), i.e, the transversal intersections of hypersurfaces of degrees d 1 , · · · , d r respectively. We call the product d 1 d 2 · · · d r the total degree, denoted by d. It is well known that all complete intersections of fixed multidegree are diffeomorphic. On the other hand, there exist diffeomorphic complete intersections with different multidegrees. For lower dimensions, such as complex dimensions 2, 3, 4, the diffeomorphic examples can be found in [1,2,8]. W. Ebeling ([3]) and A.S. Libgober-J. Wood ( [11]) independently found examples of homeomorphic complex 2-dimensional complete intersections but not diffeomorphic. In [6], F.Q. Fang and the author proved that, in dimensions n = 5, 6, 7, two complete intersections X n (d) and X n (d ) are homeomorphic if and only if they have the same total degree, Pontrjagin classes and Euler characteristics. Particularly, by Traving's result ( [7,Theorem A] or [12]), to the prime factorization of total degree d = p primes p νp(d) , if ν p (d) 2n+1 2(p−1) + 1 for all primes p with p(p−1) n+1, two homeomorphic complex n-dimensional complete intersections are diffeomorphic.
The first purpose of this paper is to give examples of diffeomorphic complex 5-dimensional complete intersections with different multidegrees. These examples, which are easy to check but hard to happen upon, were found by computer search. From these examples, we can deduce some interesting remarks about complete intersections.
Libgober and Wood ( [10]) showed the existence of homeomorphic complete intersections of dimension 2 and diffeomorphic ones of dimension 3 which belong to components of the moduli space having different dimensions. In fact it was shown that there is a procedure which allows one to produce from a pair of homeomorphic complete intersections an arbitrarily long family, all members of which are homeomorphic. P. Brückmann ([1]) shows that the construction mentioned yields families of arbitrary length t of complete intersections in CP 4t−2 (resp. CP 5t−2 ) consisting of homeomorphic complete intersections of dimension 2 (resp. diffeomorphic ones of dimension 3) but that belong to components of the moduli space of different dimensions. Furthermore, under Theorem 1 of [5], Brückmann also proves the similar result for the complete intersections of dimension 4 in CP 6t−2 ([2]).
Another purpose of this paper is to give the following theorem, which is a supplement to the results of Brückmann [1,2]. Theorem 1.1. For each integer t > 1, there exist t diffeomorphic complex 5-dimensional complete intersections in CP 7t−2 isomorphism class of which lie in different dimensional components of the moduli space.
This paper is organized as follows: After presenting the basic formulas of characteristic classes of complete intersections in Section 2, we will give examples of diffeomorphic complex 5-dimensional complete intersections in Section 3. Section 4 proves Theorem 1.1. The last section will be devoted to the code of computer program to evaluate an inequality, which is a key to prove Theorem 1.1.
Acknowledgement. This work was undertaken when the author visited the Department of Mathematical Sciences in University of Copenhagen. The author is grateful to Professor Jesper Michael Møller and Department of Mathematical Sciences for their hospitality. The author would like to thank the following students for their help on computer programming to search examples: Jianpeng Du, Mo Jia, Wenyu He, Sibo Zhao.

Characteristic classes of complete intersections
For a complete intersection X n (d), let H be the restriction of the dual bundle of the canonical line bundle over CP n+r to X n (d), and x = c 1 (H) ∈ H 2 (X n (d); Z). Associate the multidegree d = (d 1 , d 2 , . . . , d r ), define the power sums s i = r j=1 d i j for 1 i n. Then the Chern classes and Pontrjagin classes are presented as follows ( [8]): The Euler characteristic is ( e(X n (d)) = c n (X n (d)) ∩ [X n (d)] = d 1 n! g n (n + r + 1 − s 1 , . . . , n + r + 1 − s n ).

Examples of diffeomorphic complex 5-dimensional complete intersections
For complex 5-dimensional complete intersections X 5 (d 1 , . . . , d r ), its total degree, Pontrjagin classes and Euler characteristic are as follows: (3.1) Here, p 1 and p 2 denote the Pontrjagin classes as appointed in the end of Section 2. By Theorem 1.1 of [6], to find homeomorphic complex 5-dimensional complete intersections, we only need to find different multidegrees, such that (3.1)-(3.4) all agree respectively. Additionally, by [7, Theorem A], for the total degree d = p primes p νp(d) , if ν 2 (d) 7 and ν 3 (d) 4, the homeomorphic 5-dimensional complete intersections are diffeomorphic. This searching can completely be done by computer. According to [8,Proposition 7.3], let X n (d) ⊂ CP n+r be a complete intersection of given codimension r with n > 2 and 2r n + 2, then the total degree and Pontrjagin classes of X n (d) determine the multidegree. Thus, it is impossible to find out such a homeomorphic or diffeomorphic example with different multidegrees in which one of the complete intersections has codimension 2 or 3 for complex dimension 5. Theoretically, there should exist a lot of homeomorphic complete intersections with codimension 4. However, with the codimension becoming smaller, it will become more difficult to find out such examples. In fact, we can offer such examples with codimension 7 (See Section 4). The above two multidegrees have different power sums s 3 , s 5 , but they have the same total degree, Pontrjagin classes and Euler characteristic. Since d = 37362124800 = 2 11 × 3 6 × 5 2 × 7 × 11 × 13, so X 5 (66, 56, 45, 39, 16, 15, 8, 3) and X 5 (64, 60, 42, 39, 20, 11, 9, 3) are diffeomorphic.
Compare the above Examples 3.2, 3.3 and 3.4, we can obtain the following interesting remarks.
Note that, in [4], Fang asked the following question: If X n (d) and X n (d ) are diffeomorphic/or homeomorphic/or homotopy equivalent, is X n (d; a) diffeomorphic to X n (d ; a) for a natural number a? Here X n (d; a) is the complete intersection with multidegree ( d 1 , d 2 , . . . , d r , a). Now, Remark 3.5 partially gives a negative answer to Fang's question.

Moduli spaces of complete intersections
In this section, we will prove Theorem 1.1. Let X n (d) ⊂ CP N , where n 2, d = (d 1 , . . . , d r ), d i 2 and r = N − n. Then from [1, Lemma 3], the explicit formula for moduli space dimension is  We list the corresponding power sums, total degree, Pontrjagin classes and Euler characteristic in Table 1. From Table 1, the total degree is 1136843237376 = 2 11 × 3 6 × 7 × 11 2 × 29 × 31, so the two complete intersections X 5 (d) and There is a way to generate larger sets of diffeomorphic complete intersections from the above pairs d and d , which arose from [10] and had an application in [1,2].
Thus, it is clear that, with any fixed s 1, s > λ 0, m(d λ,s−λ ) form a strictly monotonously increasing sequence for λ. Hence, the Proposition follows.

Mathematica Code and outputs
In this section, Mathematica code and outputs that are designed to evaluate the inequality in Proposition 4.2 are attached in a notebook(.nb format).