On the Chern number inequalities satisfied by all smooth complete intersection threefolds with ample canonical class
aa r X i v : . [ m a t h . AG ] S e p ON THE CHERN NUMBER INEQUALITIES SATISFIED BYALL SMOOTH COMPLETE INTERSECTION THREEFOLDSWITH AMPLE CANONICAL CLASS
MAO SHENG, JINXING XU, AND MINGWEI ZHANG
Abstract.
We obtain all linear Chern number inequalities satisfied by anysmooth complete intersection threefold with ample canonical class.
Keywords : Complete intersection threefolds, Chern number inequalitiesMathematics Subject Classification 2000: 14J30 Introduction
This small note is motivated by finding a new Chern number inequality for asmooth projective threefold X with ample canonical bundle. Let c i = c i ( T X ) beits Chern class for i = 1 , ,
3. Yau’s famous inequality [4] in the three dimensionalcase says that 8 c c ≤ c , with equality iff X is uniformized by the complex ball. As it contains no c term, one may naturally wonder whether there exists a Chern number inequalityinvolving c . This is possible because of the following result: Theorem 1.1 (Chang-Lopez, Corollary 1.3 [2]) . The region described by theChern ratios ( c c c , c c c ) of smooth irreducible threefolds with ample canonical bun-dle is bounded. However, the result, as well as its proof, does not produce a new Chern numberinequality, even for the subclass of smooth complete intersections. Before the dis-covery of a new method to handle the general case, it is valuable from a scientificstandpoint to treat this subclass first by bare hands. This is what we are goingto do here.Our method is to determine the convex hull in R generated by Chern ratios( c c c , c c c ) of all smooth complete intersection threefolds with ample canonicalclass. Let n be a natural number. A smooth complete intersection (abbreviated asSCI) threefold X in P n +3 is cut out by n hypersurfaces, and a nondegenerate one,i.e., not contained in a hyperplane, by n hypersurfaces of degrees d +1 , · · · , d n +1with d i ≥ ≤ i ≤ n . The Chern numbers for a smooth X is uniquelydetermined by the tuple ( d , · · · , d n ). Therefore we may use the notation Q ( n ; d , · · · , d n ) = ( c c c , c c c ) ∈ R for Chern ratios of X . Note that X has ample canonical class if and only if P ni =1 d i ≥
5. Put Q = { Q ( n ; d , · · · , d n ) | n ≥ , d i ≥ , n X i =1 d i ≥ } ⊂ R . Let P be the convex hull of Q . Theorem 1.2. P is a rational polyhedra with infinitely many faces. The cornersof P are given by the following points: Q (1; 5) = ( , ) , Q (2; 2 ,
3) = ( , ) ,Q (3; 2 , ,
3) = ( , ) , Q (3; 2 , ,
2) = ( , ) , and Q ( n ; 1 , · · · ,
1) = ( − n ) − n + n , − n − n + n − n )(12 − n + n ) ) with n ≥ . Remark 1.3.
In another work of M.-C. Chang [1], she described a region R inthe plane of Chern ratios such that any rational point in R can be realized by aSCI threefold with ample canonical bundle; Outside R there are infinitely manyChern ratios of smooth complete intersection threefolds but no accumulatingpoints. These two results are related, but do not imply each other. See FIGURE ?? .We proceed to deduce our main application of the above result. According to thevalues of their x -coordinates, we label the corner points of P as follows: p = Q (1; 5) , p = Q (2; 2 , , p = Q (3; 2 , , ,p = Q (5; 1 , , , , , p = Q (3; 2 , , , p n = Q ( n ; 1 , · · · , , n ≥ . The sequence of points { p n } converges to the point p ∞ = (2 ,
13 ) . The closure of P , denoted by ¯ P , contains the points p n ( n ≥ , p ∞ as its corners.For two distinct points p, q ∈ R , denote the line through p, q by L pq , and the linesegment connecting p, q by pq . Denote the expressions of lines as follows: L p p ∞ : y = k x + b ,L p m p m +1 : y = k m x + b m , m ≥ . The values of k m , b m are:( k , b ) = ( − , ) , ( k , b ) = ( − , , ( k , b ) = ( − , , ( k , b ) = ( − , , ( k , b ) = ( − , ) , ( k , b ) = ( − , ,k m = − m + m + 4 m − m ( − m )( − m )( − − m + 3 m ) , ∀ m ≥ ,b m = −
120 + 254 m + 3 m − m + 9 m − m )( − m )( − − m + 3 m ) , ∀ m ≥ . The sequence of lines L p m p m +1 converges to the line L ∞ : y = k ∞ x + b ∞ , where k ∞ = − , b ∞ = 1 . HERN NUMBER INEQUALITIES 3
Theorem 1.4.
Let C be the convex cone of linear inequalities satisfied by theChern numbers of each SCI threefold with ample canonical bundle. That is, C = { ( λ , λ , λ ) ∈ R | λ c ( X ) + λ c ( X ) c ( X ) + λ c ( X ) ≥ , for any SCI threefold X with K X > . } Then C is a rational convex cone with edges ( − k , − b , , ( k m , b m , − m ≥ , ( k ∞ , b ∞ , − , where by an edge we mean a one dimensional face.Proof. Let ˇ C ⊂ R be the closure of the convex cone generated by the set { ( c ( X ) , c ( X ) c ( X ) , c ( X )) ∈ R | X is a SCI threefold with K X > . } Note that if X is a SCI threefold with K X >
0, then c ( X ) c ( X ) <
0. Indeed,Yau’s inequality gives us 8 c ( X ) c ( X ) ≤ c ( X ) , and the ampleness of canonicalbundle implies the inequality c ( X ) < c ( X ) c ( X ) <
0, it can be easily seen thatˇ C = { ( λx, λ, λy ) | λ ∈ R ≤ , ( x, y ) ∈ ¯ P } . By definition, C is the dual cone of ˇ C . By Theorem 1.2, the codimensionalone faces of ˇ C are exactly the hyperplanes in R determined by the vectors( − k , − b , k m , b m , − m ≥ k ∞ , b ∞ , − C andˇ C we get that ( − k , − b , k m , b m , − m ≥ k ∞ , b ∞ , −
1) are exactly theedges (one-dimensional faces) of C . (cid:3) Corollary 1.5. If X is a SCI threefold with K X > , then its Chern numberssatisfy the inequality c ≤ c < c , with the equality c = c holds if andonly if X is isomorphic to a degree hypersurface in P .Proof. It can be checked that744229 ( − k , − b ,
1) + 515229 ( k , b , −
1) = ( − , , . By Theorem 1.4, ( − , , ∈ C , hence we have c − c ≥ , with equality holds iff( c c c , c c c ) = L p p ∞ ∩ L p p = Q (1; 5) , which is equivalent to that X is isomorphic to a degree 6 hypersurface in P .Similarly, 93422 ( − k , − b ,
1) + 515422 ( k ∞ , b ∞ , −
1) = ( 16 , , − . By Theorem 1.4, we have 16 c − c ≥ , MAO SHENG, JINXING XU, AND MINGWEI ZHANG with equality holds iff ( c c c , c c c ) = L p p ∞ ∩ L ∞ = p ∞ . Since p ∞ is not in Q , the inequality c − c ≥ (cid:3) Remark 1.6.
In [3], the authors prove that for a smooth projective threefold X admitting a smooth fibration of minimal surfaces of general type over a curve, itholds that c ( X ) ≥ c ( X ) . According to Corollary 1.5, this inequality can never be satisfied for a SCI three-fold with ample canonical bundle. As pointed out by Professor Kang Zuo, thisis actually explained by the Lefschetz hyperplane theorem. Indeed, the hyper-bolicity of the moduli space of minimal surfaces of general type means that thebase curve of a smooth fibration is a hyperbolic curve and hence the fundamentalgroup of the total space X is nontrivial, in contrast with the triviality of thefundamental group of a SCI implied by the Lefschetz hyperplane theorem.2. Proof of the Theorem
Suppose n ∈ N is a positive integer, and d , · · · , d n ∈ R ≥ are nonnegative realnumbers. Let s j = P ni =1 d ji , j ≥
1. We define c ( n ; d , · · · , d n ) := 4 − s ,c ( n ; d , · · · , d n ) := s + s − s − ,c ( n ; d , · · · , d n ) := − s + 3 s s + 2 s s + s ) − s + 4 . If d = · · · = d n = d , we denote c i ( n ; d , · · · , d n ) by c i ( n ; d ), i = 1 , , . The following result is standard.
Lemma 2.1.
Let X be a SCI threefold in P n +3 . If X is a complete intersectionof hypersurfaces with degrees d + 1 , · · · , d n + 1 , and d i ≥ , ∀ ≤ i ≤ n , thenthe Chern classes of X are: c i ( X ) = c i ( n ; d , · · · , d n ) , i = 1 , , . We divide the proof of Theorem 1.2 into three steps, corresponding to threesections.Step 1: In section 2.1, we firstly prove the x -coordinate of any point in Q isbetween the x -coordinates of p and p ∞ . Then we prove any point of Q is belowthe line L p p ∞ .Step 2: In section 2.2, we prove that any point of Q is above the line L p m p m +1 , ∀ m ≥ ∀ i = 1 , · · · ,
5, if a point Q ( n ; d , · · · , d n ) ∈ Q has x -coordinate less than or equal to the x -coordinate of p , then Q ( n ; d , · · · , d n )lies above the line segment p i p i +1 .After the three steps above, it is obviously we have finished the proof of Theorem1.2. HERN NUMBER INEQUALITIES 5
Ideas of the proof in each step:In steps 1 and 2, the idea of the proof is the following:Given a line L : y = kx + b in R , to prove Q is below L is equivalent to verify ∀ n, d i ∈ N , P ni =1 d i ≥ ,c ( n ; d , · · · , d n ) − kc ( n ; d , · · · , d n ) − bc ( n ; d , · · · , d n ) c ( n ; d , · · · , d n ) ≥ , and by the following Lemma 2.2, it suffices to verify ∀ n ∈ N , d ∈ R , d ≥ , nd ∈ N , nd ≥ c ( n ; d ) − kc ( n ; d ) − bc ( n ; d ) c ( n ; d ) ≥ . Similarly, in order to prove Q is above a line y = kx + b , it suffices to verify ∀ n ∈ N , d ∈ R , d ≥ , nd ∈ N , nd ≥ kc ( n ; d ) + bc ( n ; d ) c ( n ; d ) − c ( n ; d ) ≥ . Lemma 2.2.
Let λ, µ, ν ∈ R be constants. ∀ m ∈ N , we have inf { λc ( n ; d , · · · , d n ) + µc ( n ; d , · · · , d n ) c ( n ; d , · · · , d n ) + νc ( n ; d , · · · , d n ) | n ∈ N , d i ∈ R , n X i =1 d i = m, d i ≥ , ∀ i = 1 , · · · , n. } ≥ inf { λc ( n ; d ) + µc ( n ; d ) c ( n ; d ) + νc ( n ; d ) | n ∈ N , d ∈ R , nd = m, d ≥ . } Proof.
This lemma is a direct consequence of the following elementary proposi-tion. (cid:3)
Proposition 2.3.
Let d ≤ d ≤ · · · ≤ d n be nonnegative real numbers, s j = P ni =1 d ji , j = 1 , , , and λ, µ ∈ R be constants. For fixed n and s , there exists anatural number k ≤ n , such that the function λs + µs attains its minimal valuewhen d = · · · = d k = 0 , and d k +1 = · · · = d n = s n − k . In step 3, we firstly prove that there are only finite points of Q with x -coordinatesless than or equal to the x -coordinate of p , then we verify case-by-case that if apoint of Q has x -coordinate less than or equal to the x -coordinate of p , it liesabove the union of line segments ∪ i =1 p i p i +1 .2.1. We first give an estimate of the x -coordiantes of points in Q . Lemma 2.4.
The x -coordinate of any point of Q is between the x -coordiantes of p and p ∞ .Proof. Recall the x -coordinates of p and p ∞ are and 2 respectively. For anypoint Q ( n ; d , · · · , d n ) in Q , by Lemma 2.1, its x -coordinate is c ( n ; d , · · · , d n ) c ( n ; d , · · · , d n ) c ( n ; d , · · · , d n ) = 2(4 − s ) s + s − s − s j = P ni =1 d i , j = 1 , ≤ − s ) s + s − s − ≤ s ≥
5, the inequalities above can be verified easily. (cid:3)
MAO SHENG, JINXING XU, AND MINGWEI ZHANG
Recall the line L p p ∞ has the expression y = k x + b , where k = − , b = .According to the argument before Lemma 2.2, in order to prove Q is below L p p ∞ ,it suffices to prove the following : Lemma 2.5. ∀ n ∈ N , d ∈ R , nd ∈ N , nd ≥ , d ≥ ,f ( n, d ) := c ( n ; d ) − k c ( n ; d ) − b c ( n ; d ) c ( n ; d ) ≥ . Proof.
Let s = nd , we have f ( n, d ) = ˜ f ( s , d ) = 193 (3500 − s − ds − d s + 422 s + 211 ds ) . Since ˜ f ( s , d ) is a quadratic polynomial of d with negative leading term, and1 ≤ d ≤ s , we have ˜ f ( s , d ) ≥ M in { ˜ f ( s , , ˜ f ( s , s ) } . By computations, f ( s ,
1) = (3500 − s + 633 s ), f ( s , s ) = (700 − s − s + 36 s ).It is elementary to verify the above two polynomials of s are nonnegative when s ∈ N and s ≥
5. Hence f ( n, d ) = ˜ f ( s , d ) ≥ , ∀ n ∈ N , d ∈ R , nd ∈ N , nd ≥ , d ≥ . (cid:3) Q is above the line L p m p m +1 , ∀ m ≥
6. Recallthe line L p m p m +1 has an expression y = k m x + b m , where k m = − m + m + 4 m − m ( − m )( − m )( − − m + 3 m ) ,b m = −
120 + 254 m + 3 m − m + 9 m − m )( − m )( − − m + 3 m ) . According to the argument before Lemma 2.2, to prove Q is above the line L p m p m +1 , we need to study the nonnegativity of the function f ( m, n, d ) := k m c ( n ; d ) + b m c ( n ; d ) c ( n ; d ) − c ( n ; d ) . We have the following lemma.
Lemma 2.6. f ( m, n, d ) ≥ , if one of the following conditions holds: (1) m, n ∈ N , d ∈ R , d ≥ , nd ∈ N , nd ≥ , m ≥ ; (2) m, n ∈ N , d ∈ R , d ≥ , nd ∈ N , nd ≥ , m = 6 , , , .Proof. Let s = nd . In the new variable m, s , d , we denote the function f ( m, n, d )by ˜ f ( m, s , d ), then by the expressions of k m , b m and c i ( n ; d ),˜ f ( m, s , d ) = f ( m, n, d )= k m (4 − s ) + b m (4 − s )( s − s + 6) + s − s + 3 s − s d s − s + b m (4 − s ) s d. Suppose condition (1) holds, since ˜ f ( m, s , d ) is a quadratic polynomial of d , wehave two cases: HERN NUMBER INEQUALITIES 7
Case I: − ( s / − s + b m (4 − s ) s / s / ≤ . In this case, ˜ f ( m, s , d ) ≥ ˜ f ( m, s ,
1) = f ( m, s , f ( m, s , ≥ Q ( s ; 1 , · · · ,
1) lies above the line L p m p m +1 , which can beeasily verified under the condition s ≥ − ( s / − s + b m (4 − s ) s / s / ≥ . In this case, s ≥ b m − b m − , and˜ f ( m, s , d ) ≥ ˜ f ( m, s , − ( s / − s + b m (4 − s ) s / s / . By computations, we get g ( m, s ) :=12( − m ) ( − m ) ( − − m + 3 m ) · ˜ f ( m, s , − ( s / − s + b m (4 − s ) s / s / s with polynomial coefficients of m . We only need toverify the positivity of g ( m, s ).Again, by computations, we get that if m ∈ N , m ≥
10, then g ( m, b m − b m − > , ∂g∂s ( m, b m − b m − > ,∂ g∂s ( m, b m − b m − > , ∂ g∂s ( m, b m − b m − > . Note g ( m, s ) is a cubic polynomial of s , the positivity of g ( m, s ) follows fromthe above computations, and we have verified the conclusion under condition (1).Suppose condition (2) holds, by computations, we have12 b m − b m − < , ∀ m = 6 , , , . This inequality and condition (2) imply s > b m − b m − + 1 > b m − b m − . We have˜ f ( m, s , d ) ≥ ˜ f ( m, s , − ( s / − s + b m (4 − s ) s / s / . Again let g ( m, s ) :=12( − m ) ( − m ) ( − − m + 3 m ) · ˜ f ( m, s , − ( s / − s + b m (4 − s ) s / s / MAO SHENG, JINXING XU, AND MINGWEI ZHANG which is a cubic polynomial of s . We only need to show the positivity of g ( m, s ).By direct computations, g ( m, b m − b m − > , ∂g∂s ( m, b m − b m − > ,∂ g∂s ( m, b m − b m − > , ∂ g∂s ( m, b m − b m − > . Since g ( m, s ) is a cubic polynomial of s , the positivity of g ( m, s ) follows fromthe above computations. And we have verified the conclusion under condition(2). (cid:3) From Lemma 2.2 and Lemma 2.6, we get that Q ( n ; d , · · · , d n ) ∈ Q lies abovethe line L p m p m +1 , if one of the following conditions holds:(1) m ≥ m = 6 , , , , P ni =1 d i ≥ ≤ P ni =1 d i ≤
10, then Q ( n ; d , · · · , d n )lies above the line L p m p m +1 , ∀ m = 6 , , ,
9. So we have verified Q lies above theline L p m p m +1 , ∀ m ≥ ∀ i = 1 , · · · ,
5, if a point Q ( n ; d , · · · , d n ) ∈ Q has x -coordinate less than or equal to the x -coordinate of p , then Q ( n ; d , · · · , d n )lies above the line segment p i p i +1 .Note that the x -coordinate of p is . The following lemma tells us that there areonly finite points in Q with x -coordinate less than or equal to the x -coordinateof p . Lemma 2.7. If s = P ni =1 d i ≥ , then c ( n ; d , · · · , d n ) c ( n ; d , · · · , d n ) c ( n ; d , · · · , d n ) = (4 − s ) ( s + s ) / − s − > . Proof.
Since s ≤ s , we have s ≥ ⇒ s − s + 24 > ⇒ s − s + 120 > ⇒ s − s + 120 > s ⇒ s − s + 120 > s ⇒ (4 − s ) ( s + s ) / − s − > . (cid:3) A case-by-case verification shows that the finite points { Q ( n, d , · · · , d n ) , ≤ P ni =1 d i ≤ } lie above the lines { L p i p i +1 , ≤ i ≤ } . By these verificationsand the above lemma, we have shown that, once the x -coordinate of a point Q ( n ; d , · · · , d n ) is less than or equal to that of p , it lies above the line L p i p i +1 , ∀ ≤ i ≤
5. This completes the proof of Theorem 1.2.
HERN NUMBER INEQUALITIES 9
Acknowledgments
The authors would like to express warm thanks to Professor Sheng-Li Tan andProfessor Kang Zuo for helpful discussions and comments. We would like alsoto thank Professor Ulf Persson for his comments, especially sharing with us hisconjecture on the geography of the Chern invariants of threefolds. This work ispartially supported by Wu Wen-Tsun Key Laboratory of Mathematics, Universityof Science and Technology of China.
References [1] Chang, M.-C.;
Distributions of Chern numbers of complete intersection threefolds .Geom. Funct. Anal. 7 (1997), 861-872.[2] Chang, M.-C.; Lopez, A.-F.;
A linear bound on the Euler number of threefolds ofCalabi-Yau and of general type . Manuscripta Math. 105 (2001), 47-67.[3] Lu, J.; Tan, S.-L.; Zuo, K.;
Canonical class inequality for fibred spaces .arXiv:1009.5246v2.[4] Yau, S.-T.;
Calabi’s conjecture and some new results in algebraic geometry . Proc. Nat.Acad. Sci. U.S.A. 74 (1977), no. 5, 1798-1799.
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