Endomorphisms of hypersurfaces of Fano manifolds of Picard number 1
aa r X i v : . [ m a t h . AG ] J u l ENDOMORPHISMS OF HYPERSURFACES OF FANOMANIFOLDS OF PICARD NUMBER 1
INSONG CHOE
Abstract.
It is conjectured that a Fano manifold X of Picard number1 which is not a projective space admits no endomorphisms of degreebigger than 1. Beauville confirmed this for hypersurfaces of projectivespace. We study this problem when X is given by a hypersurface of anarbitrary Fano manifold of Picard number 1. In this paper, we are concerned with the following conjecture on theendomorphisms of Fano manifolds of Picard number 1.
Conjecture 0.1. ([AKP], Conjecture 1.1) Let f be a surjective endomor-phism of a Fano manifold X of Picard number 1. If deg f >
1, then X ∼ = P n .The reader is referred to the paper [AKP] of Aprodu, Kebekus, and Peternellfor known results on this conjecture and their recent enlargement of thelist of Fano manifolds giving evidence to this conjecture. Also one can seeFakhruddin’s paper [Fak] for the arithmetic features of this problem.On projective hypersurfaces, Beauville proved the following. Theorem 0.2. ([B]) A smooth complex projective hypersurface of dimen-sion bigger than 1 and degree bigger than 2 admits no endomorphisms ofdegree bigger than 1.In particular, this shows that the projective hypersurfaces of low degree,which are Fano manifolds of Picard number 1, satisfy the conjectured prop-erty.The proof of this result in [B] relies mainly on the following Chern num-ber inequality which is obtained from the Hurwitz type formula devised byAmerik, Rovinsky and Van de Ven.
Lemma 0.3. ([ARV], Corollary 1.2) Let f : X → Y be a finite morphismbetween projective varieties of dimension n . Let L be a line bundle on Y such that Ω Y ( L ) is globally generated. Then(0.1) deg( f ) · c n (Ω Y ( L )) ≤ c n (Ω X ( f ∗ L )) . To get the wanted result, Beauville combined this formula with explicitcomputations of the Chern numbers of projective hypersurfaces.
Mathematics Subject Classification.
Primary: 14J70, 14M20.
Key words and phrases.
Fano manifold, endomorphism, hypersurface, Hurwitz typeformula.
The aim of this paper is to apply the formula (0.1) in a much moregeneral context. The main idea is to replace the explicit Chern numbercomputation in [B] by another simple Chern number inequality. Our resultis the following.
Theorem 0.4.
Let V be a Fano manifold with P ic ( V ) ∼ = Z generated by O V (1) . Assume that dim( V ) ≥ and Ω V ( l ) is globally generated. Let X bea smooth hypersurface of V cut out by a member of |O V ( d ) | , d ≥ l . Then X admits no surjective endomorphisms of degree bigger than 1. An interesting special case is when V is a prime Fano manifold, whichmeans by definition that the ample generator O V (1) is very ample. Forexample, compact Hermitian symmetric spaces and homogeneous contactmanifolds of Picard number 1 are known to be prime. In this case, V isembedded in a projective space P N via O V (1) and thus Ω V (2) is globallygenerated, because it is a quotient of a globally generated bundle Ω P N (2).Hence we have the following. Recall that the index of V is defined as thenumber i satisfying K V ∼ = O V ( − i ). Theorem 0.5.
Let V be a prime Fano manifold of Picard number 1 withindex i . Assume that dim V ≥ . Let X be a smooth hypersurface of V cutout by a member of |O V ( d ) | , ≤ d < i . Then X is a Fano manifold ofPicard number 1 and X admits no surjective endomorphisms of degree > .Proof. Non-existence of endomorphisms is a consequence of the above the-orem. So it suffices to check that X is a Fano manifold of Picard number1. From the Fano condition on V , H ( V, O V ) = H ( V, O V ) = 0. By Lef-schetz hyperplane theorem, the restriction of divisors yields the isomorphism P ic ( X ) ∼ = P ic ( V ) ∼ = Z . The Fano condition on X can be checked by theadjunction formula K X ∼ = K V | X ⊗ [ X ] | X ∼ = O X ( − i + d ) , where O X (1) = O V (1) | X is the ample generator of P ic ( X ). (cid:3) Proof
In this section, we prove Theorem 0.4. Suppose that X and V satisfy theassumptions of Theorem 0.4. Let dim X = n ≥
3, let h = c ( O X (1)). Firstwe have an upper bound on a Chern number. Lemma 1.1. (cf. [B], Proposition 1) If X has a surjective endomorphismof degree bigger than 1, then c n (Ω X ( l )) ≤ l n · h n . Proof.
Let f be an endomorphism of degree bigger than 1. Let m > f ∗ O X (1) ∼ = O X ( m ). Since Ω V ( l ) is globally generated,Ω X ( l ) is also globally generated. Hence by Lemma 0.3, we have the followinginequality(1.1) m n · c n (Ω X ( l )) ≤ c n (Ω X ( lm )) . NDOMORPHISMS OF HYPERSURFACES 3
Expanding c n (Ω X ( lm )) as a polynomial of m , we get c n (Ω X ( lm )) = l n · h n · m n + (lower order terms) . By iterating the endomorphism f , we can make m arbitrarily large. Hencedividing both sides of (1.1) by m n and taking limit m → ∞ , we get thewanted inequality. (cid:3) Next from the conormal sequence twisted by O X ( l ):0 → O X ( l − d ) → Ω V ( l ) | X → Ω X ( l ) → , we get c (Ω X ( l )) = c (Ω V ( l ) | X ) · (1 + ( l − d ) h ) − . Hence the top Chern number has the following expression:(1.2) c n (Ω X ( l )) = n X i =0 ( d − l ) i · c n − i (Ω V ( l ) | X ) h i . Since Ω V ( l ) is globally generated, its restriction to X is also globally gener-ated. Hence the Chern classes c i (Ω V ( l ) | X ) are represented by positive cyclesof pure codimension i (cf. [Ful], Example 14.4.3). Hence for d ≥ l , each termappearing in the summation (1.2) is nonnegative. In particular, we have Lemma 1.2.
For d ≥ l , c n (Ω X ( l )) ≥ ( d − l ) n h n . (cid:3) To finish the proof of Theorem 0.4, we compare the bounds on c n (Ω X ( l ))given by Lemma 1.1 and Lemma 1.2 and get d ≤ l .For d = 2 l , from Lemma 1.1 and (1.2), c n (Ω X ( l )) = n X i =0 l i · c n − i (Ω V ( l ) | X ) h i ≤ l n h n . Because c n − i (Ω V ( l ) | X ) · h i ≥
0, we get c n (Ω X ( l )) = l n h n and c n − i (Ω V ( l ) | X ) · h i = 0for each i = 0 , , · · · , n −
1. But this is absurd: from the vanishing c (Ω V ( l ) | X ) = c ( K V ⊗ O V ( l ( n + 1)) | X ) = 0, we get K V ∼ = O V ( − ln − l ). From the well-known upper bound on the index of Fano manifolds of Picard number 1([KO] p.32), we have ln + l ≤ n + 2. Thus l = 1 and K V ∼ = O V ( − n − V is a hyperquadric ([Fuj] Theorem 2.3 or [BS] 3.1.6).But in this case it is known that H ( V, Ω V (1)) = 0 ([S], Section 4), contraryto the assumption that Ω V ( l ) is globally generated.Therefore, we see that an existence of surjective endomorphism of degreebigger than 1 yields d < l . (cid:3) Remark 1.3.
The above method doesn’t work for the case d ≤ l . For l < d < l (for d = 3 when V is prime), it is still possible that the inequalityin Lemma 1.1 yields a contradiction. For example, the inequality is violatedfor cubic projective hypersurfaces ([B]). Also when V is the Grassmannian Gr (2 ,
4) and X is a smooth hypersurface with O V ( X ) ∼ = O V (3), direct INSONG CHOE computation shows that c (Ω X (2)) = 876 which is bigger than the expectedupper bound 2 · h . Remark 1.4.
In the above proof, the Fano condition on V was not essen-tially used. In the proof, we needed the Fano condition only to guaranteethat P ic ( X ) ∼ = Z and it is generated by the hyperplane section. Thus,the Fano condition on V in Theorem 0.4 can be replaced by the followingcohomological conditions: H ( V, O V ) = H ( V, O V ) = 0 . Complete intersections
In this section, we find a direct generalization of Theorem 0.4 to the caseof complete intersections. Here we just consider the case when V is prime. Theorem 2.1.
Let V be a prime Fano manifold of Picard number 1. Let ( d i ) ≤ i ≤ k be positive numbers and let X = V ∩ H ∩ H ∩ · · · ∩ H k be a smooth subvariety of V of codimension k cut out by general hypersur-faces H i ∈ |O V ( d i ) | . Assume that dim X ≥ . If d i ≥ for some i , then X admits no surjective endomorphisms of degree > .Proof. Since H i is general, we have a smooth variety X i := V ∩ H ∩ · · · ∩ H i for each i . By Lefschetz hyperplane theorem, P ic ( X i ) ∼ = Z and it is gener-ated by O V (1) | X i . Rearranging the sequence of hypersurfaces { H i } , we mayassume that d k ≥
4. The smooth variety X k − satisfies all the conditions of V in Theorem 0.4, possibly except for the Fano condition. Also, X = X k isa smooth hypersurface of X k − of degree d k ≥
4. Hence by Remark 1.4, X admits no surjective endomorphisms. (cid:3) Acknowledgements.
We would like to thank J.-M. Hwang for readingthe first draft of this paper and giving a remark on the prime conditionin Theorem 0.5. Also we would like to thank the referee for informing thereferences [BS] and [Ful].
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Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-Gu, Seoul, Korea
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