Miyaoka-Yau type inequalities of complete intersection threefolds in products of projective spaces

. Geography of projective varieties is one of the fundamental problems in algebraic geometry. There are many researches toward the characteristics of Chern number of some projective spaces, for example Noether’s inequalities, the theorem of Chang-Lopez, and the Miyaoka-Yau inequality. In this paper, we compute the Chern numbers of any smooth complete intersection threefold in the product of projective spaces via the standard exact sequences of cotangent bundles. Then we obtain linear Chern number inequalities for 𝑐 1 (𝑋)𝑐 2 (𝑋) 𝑐 13 (𝑋) and 𝑐 3 (𝑋) 𝑐 13 (𝑋) on such threefolds under conditions of 𝑑 𝑖𝑗 ≥ 4 and 𝑑 𝑖𝑗 ≥ 6 respectively . They can be considered as a generalization of the Miyaoka-Yau inequality and an improvement of Yau’s inequality for such threefolds.


Introduction
One of the fundamental problems in algebraic geometry is to study the geogra-phy of projective varieties, i.e., determining which Chern numbers occur for a complex smooth projective variety M. When M is a minimal surface of general type, we have Noether's inequalities [1]: This implies 5 1 2 () ≥  2 () − 36.
The theorem of Chang-Lopez has been generalized to higher dimensional case by Du and Sun in [6].
Theorem 1.1.Let X be a nonsingular projective variety of dimension n over an algebraic closed field κ with any characteristic.Suppose KX or −KX is ample.If the characteristic of κ is 0 or the characteristic of κ is positive and OX(KX)(OX(−KX), respectively) is globally generated, then is contained in a convex polyhedron in A p(n) depending on the dimension of X only, where p(n) is the partition number and the elements in the parentheses arranged from small to big in terms of the alphabet order of the lower indices of the numerators.
In this paper, we study the inequalities of Chern numbers of complete intersection threefolds in products of ℙ 1 .Throughout this paper, we always let   : ℙ 1 × ℙ 1 × ⋯ × ℙ 1 ⏟ +3copies → ℙ 1 be the i-th projection, and   =   * () , where  is a point of ℙ 1 .Take Hi be a general divisor in the linear system | ∑ +3 =1     |, where dit is a positive integer for 1 ≤ i ≤ n,1 ≤ t ≤ n+3.By the Bertini theorem, one can assume that Hi is a smooth hypersurface for i = 1, 2, • • • , n, and  =  1 ∩  2 ∩ ⋯ ∩   is a smooth threefold.

Chern classes
In this section, we introduce the definition of Chern classes.
Let M be a smooth projective variety of dimension n.Let () =⊕ =1    () be the Chow ring of M. E is a vector bundle on M of rank r.The Chern class   () is a cycle in   (), here c0(E) = 1.We let   () = 1 +  1 () + ⋯ +   ()  be the Chern polynomial of E.
(4) Let s be a global section of E. Assume that the zero set Z(s) of s satisfies that dim Z(s) =dimM − r, then   () = () ∈   ().
We call   () =   (  ) the i-th Chern class of M.

Chern numbers of complete intersection three-folds in products of projective spaces
In this section, we compute the Chern numbers of X.
From the standard exact sequence after taking duality, we have Hence we have From the exact sequence By repeating the procedure above, it can be obtained that It follows that By considering the coefficient of t, we can get As for the coefficient of t 2 , we see that Simple computations show that Hence we have Now considering the coefficient of t where  1 , ⋯ ,   , , ,  take all the arrangements of 1,2, ⋯ ,  + 3.By (1), ( 2), (3), we can have

Inequalities of Chern numbers
In this section, we estimate the upper and lower bounds for and  3 ()  1 3 () respectively.Let We have In order to estimate , we need to estimate Proof.The desired conclusion follows from Lemma 2.

Inequalities of
In order to estimate the range of , we need to estimate the range of +    +2  )(  +2)−  (12)

𝑛+3
as an example to calculate the Chern numbers of complete intersection three-folds in products of projective spaces.Thus, in our conclusion, we get its Chern number and the inequalities that it will satisfy: If   ≥ 4 for any 1 ≤  ≤ , 1 ≤  ≤  + 3, then we have However, those conclusions build up on an important assumption, which is the value of   .This means that there is still room for exploration and explanation of those results when applying other values of   .
As for the future meaning of research into this field, it may help in the field of physics.For instance, Miyaoka-Yau type inequalities are widely applied to the quantum mechanics and field theory, so we believe researches like this can be applied to more different conditions.
If d  ≥ 6 for  , , ℎ  ℎ       >   .In order to see the relationship between       and   , we need to cal-culate the value of       >   .If   ≥ 6 for any i, j, then we have   <     ,   <     .
Theorem 4.2.If   ≥ 6 for any i, j, then we have