arxiv: v1 [math.ag] 13 Mar 2019


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1 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv: v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show that any n dimensional Hodge diamond with values in Z/mZ is attained by the Hodge numbers of an ndimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of ndimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Kollár in Introduction Hodge theory allows to decompose the kth Betti cohomology of an ndimensional Kähler manifold X into its (p, q)pieces for all 0 k 2n: H k (X,C) = H p,q (X), H p,q (X) = H q,p (X). p+q=k 0 p,q n The Clinear subspaces H p,q (X) are naturally isomorphic to the Dolbeault cohomology groups H q (X, Ω p X ). The integers h p,q (X) = dim C H p,q (X) for 0 p, q n are called Hodge numbers. One usually arranges them in the so called Hodge diamond: h n,n h n,n 1 h n 1,n h n,1 h 1,n h n,0 h n 1,1 h 1,n 1 h 0,n h n 1, h 1,0 h 0,1 h 0,0 h 0,n 1 The sum of the kth row of the Hodge diamond equals the kth Betti number. We always assume that a Kähler manifold is compact and connected, so we have h 0,0 = h n,n = 1. Date: March 13, Mathematics Subject Classification. 32Q15, 14C30, 14E99, 51M15. Key words and phrases. Hodge numbers, Kähler manifolds, construction problem. This work is supported by the DFG project Topologische Eigenschaften von algebraischen Varietäten. 1
2 2 MATTHIAS PAULSEN AND STEFAN SCHREIEDER Complex conjugation and the hard Lefschetz theorem induce the symmetries h p,q = h q,p = h n p,n q for all 0 p, q n. Additionally, we have the Lefschetz inequalities h p,q h p+1,q+1 for p + q < n. While Hodge theory places severe restrictions on the geometry and topology of Kähler manifolds, Simpson points out in [Sim04] that very little is known to which extent the theoretically possible phenomena actually occur and how far arbitrary Kähler manifolds differ in their behaviour from the strictly smaller class of smooth complex projective varieties. The real abundance of the class of Kähler manifolds is thus far from being understood. A natural question in this context is whether the above restrictions on Hodge diamonds are the only ones. This leads to the following construction problem for Hodge numbers: Question 1. Let (h p,q ) 0 p,q n be a collection of nonnegative integers with h 0,0 = 1 obeying the Hodge symmetries h p,q = h q,p = h n p,n q for 0 p, q n and the Lefschetz inequalities h p,q h p+1,q+1 for p + q < n. Does there exist a Kähler manifold X such that h p,q (X) = h p,q for all 0 p, q n? After results in dimensions two and three (see e. g. [Hun89]), significant progress has been made by the second author [Sch15] in arbitrary dimensions. For instance, it is shown in [Sch15] that the above construction problem is fully solvable for ndimensional smooth complex projective varieties if, for given k N, one assumes h p,0 = 0 for all 1 p n 1 with p k, one ignores the Hodge numbers h p,q with p + q = n, and one imposes a lower bound of magnitude pn 2 on h p,p h p 1,p 1 for 1 p < n 2. In particular, the Hodge numbers in a given weight k may be arbitrary (up to a quadratic lower bound on h p,p if k = 2p is even) and so the outer Hodge numbers can be far larger than the inner Hodge numbers, contradicting earlier expectations formulated in [Sim04]. Weaker results with simpler proofs, concerning the possible Hodge numbers in a given weight, have later been obtained by Arapura [Ara16]. In [Sch15], it was also observed that one cannot expect a positive answer to Question 1 in its entirety. For example, any 3dimensional Kähler manifold X with h 1,1 (X) = 1 and h 2,0 (X) 1 satisfies h 2,1 (X) < 12 6 h 3,0 (X), see [Sch15, Proposition 28]. Therefore, a complete classification of all possible Hodge diamonds of Kähler manifolds or smooth complex projective varieties seems out of reach. While these inequalities aggravate the construction problem for Hodge numbers, one might ask whether there also exist number theoretic obstructions for possible Hodge diamonds. For example, the Chern numbers of Kähler manifolds satisfy certain congruences due to integrality conditions implied by the Hirzebruch Riemann Roch theorem. For an arbitrary integer m 2, let us consider the Hodge numbers of a Kähler manifold in Z/mZ, which forces all inequalities to disappear. The purpose of this paper is to show that Question 1 is modulo m completely solvable even for smooth complex projective varieties: Theorem 2. Let m 2 be an integer. For any integer n 1 and any collection of integers (h p,q ) 0 p,q n such that h 0,0 = 1 and h p,q = h q,p = h n p,n q for 0 p, q n,
3 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER 3 there exists a smooth complex projective variety X of dimension n such that for all 0 p, q n. h p,q (X) h p,q (mod m) Therefore, the Hodge numbers of Kähler manifolds do not follow any number theoretic rules, and the behaviour of smooth complex projective varieties is the same in this aspect. As a consequence of Theorem 2, we show that there are no polynomial relations among the Hodge numbers (not reduced modulo m anymore) of ndimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, see Corollary 8 below. In particular, there are no such polynomial relations in the strictly larger class of Kähler manifolds, which was a question raised by Kollár after a colloquium talk of Kotschick at the University of Utah in fall For linear relations among Hodge numbers, this question was settled in work of Kotschick and the second author in [KS13]. We call the Hodge numbers h p,q (X) with p {0, n} or q {0, n} (i. e. the ones placed on the border of the Hodge diamond) the outer Hodge numbers of X and the remaining ones the inner Hodge numbers. Note that the outer Hodge numbers are birational invariants and are thus determined by the birational equivalence class of X. Our proof shows (see Theorem 4 below) that any smooth complex projective variety is birational to a smooth complex projective variety with prescribed inner Hodge numbers in Z/mZ. As a corollary, there are no polynomial relations among the inner Hodge numbers within a given birational equivalence class. This is again a generalization of the corresponding result for linear relations obtained in [KS13, Theorem 2]. The proof of Theorem 2 can thus be divided into two steps: First we solve the construction problem modulo m for the outer Hodge numbers. This is done in Section 2. Then we show the aforementioned result that the inner Hodge numbers can be adjusted arbitrarily in Z/mZ via birational equivalences (in fact, via repeated blowups). This is done in Section 3. Finally, in Section 4 we deduce that no nontrivial polynomial relations between Hodge numbers exist, thus answering Kollár s question. 2. Outer Hodge numbers We prove the following statement via induction on the dimension n 1. Proposition 3. For any collection of integers (h p,0 ) 1 p n, there exists a smooth complex projective variety X n of dimension n together with a very ample line bundle L n on X n such that h p,0 (X n ) h p,0 (mod m) for all 1 p n and χ(l 1 n ) 1 (mod m). Proof. We take X 1 to be a curve of genus g where g h 1,0 (mod m). Further, we take L 1 to be a line bundle of degree d on X 1 where d > 2g and d g (mod m). Then L 1 is very ample and by the Riemann Roch theorem we have χ(l 1 1 ) 1 (mod m). Now let n > 1. We define a collection of integers (k p,0 ) 1 p n 1 recursively via k 1,0 = 0, k 0,0 = 1, k p,0 = h p,0 2k p 1,0 k p 2,0 for 1 p n 1.
4 4 MATTHIAS PAULSEN AND STEFAN SCHREIEDER We choose X n 1 and L n 1 by induction hypothesis such that h p,0 (X n 1 ) k p,0 (mod m) for all 1 p n 1. Let E be a smooth elliptic curve and let L be a very ample line bundle of degree d on E such that d 1 (mod m). Let e be a positive integer such that n e 1 + ( 1) p h p,0 (mod m). p=1 Let X n X n 1 E E be a hypersurface defined by a general section of the very ample line bundle P n = pr 1 L n 1 pr 2 Lm 1 pr 3 Le on X n 1 E E. By Bertini s theorem, we may assume X n to be smooth and irreducible. Let L n be the restriction to X n of the very ample line bundle Q n = pr 1L n 1 pr 2L pr 3L on X n 1 E E. Then L n is again very ample. By the Lefschetz hyperplane theorem, we have h p,0 (X n ) = h p,0 (X n 1 E E) for all 1 p n 1. Since the Hodge diamond of E E is Künneth s formula yields , h p,0 (X n ) = h p,0 (X n 1 ) + 2h p 1,0 (X n 1 ) + h p 2,0 (X n 1 ) k p,0 + 2k p 1,0 + k p 2,0 (mod m) = h p,0 for all 1 p n 1. Therefore, it only remains to show that h n,0 (X n ) h n,0 (mod m) and χ(l 1 n ) 1 (mod m). Since n χ(o Xn ) = 1 + ( 1) p h p,0 (X n ), p=1 the congruence h n,0 (X n ) h n,0 (mod m) is equivalent to χ(o Xn ) e (mod m). By definition of X n, the ideal sheaf on X n 1 E E of regular functions vanishing on X n is isomorphic to the sheaf of sections of the dual line bundle Pn 1. Hence, there is a short exact sequence (1) 0 P 1 n O Xn 1 E E i O Xn 0
5 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER 5 of sheaves on X n 1 E E where i: X n X n 1 E E denotes the inclusion. Together with Künneth s formula and the Riemann Roch theorem, we obtain Tensoring (1) with Q 1 n and thus χ(o Xn ) = χ(o Xn 1 E E) χ(p 1 n ) = χ(o Xn 1 ) χ(o E ) 2 χ(l 1 n 1 }{{} ) χ(l 1 m ) χ(l e ) }{{}}{{}}{{} =0 1 1 e e (mod m). yields the short exact sequence 0 P 1 n Q 1 n Q 1 n χ(l 1 n ) = χ(q 1 n ) χ(pn 1 Q 1 n ) This finishes the induction step. i i Q 1 n 0 = χ(l 1 n 1 }{{} ) χ(l 1 ) 2 χ(l 2 n 1 }{{} ) χ(l m ) χ(l e 1 ) }{{} (mod m). 3. Inner Hodge numbers We now show the following result, which significantly improves [KS13, Theorem 2]. Theorem 4. Let X be a smooth complex projective variety of dimension n and let (h p,q ) 1 p,q n 1 be any collection of integers such that h p,q = h q,p = h n p,n q for 1 p, q n 1. Then X is birational to a smooth complex projective variety X such that for all 1 p, q n 1. h p,q (X ) h p,q (mod m) Together with Proposition 3, this will complete the proof of Theorem 2. Let us recall the following result on blowups, see e. g. [Voi03, Theorem 7.31]: If X denotes the blowup of a Kähler manifold X along a closed submanifold Z X of codimension c, we have Therefore, c 1 H p,q ( X) = H p,q (X) H p i,q i (Z). (2) h p,q ( X) c 1 = h p,q (X) + h p i,q i (Z). In order to prove Theorem 4, we first show that we may assume that X contains certain subvarieties, without modifying its Hodge numbers modulo m. Lemma 5. Let X be a smooth complex projective variety of dimension n. Let r, s 0 be integers such that r + s n 1. Then X is birational to a smooth complex projective variety X of dimension n such that h p,q (X ) h p,q (X) (mod m) for all 0 p, q n and such that X contains at least m disjoint smooth closed subvarieties that are all isomorphic to a projective bundle of rank r over P s. i=1 i=1
6 6 MATTHIAS PAULSEN AND STEFAN SCHREIEDER Proof. We first blow up X in a point and denote the result by X. The exceptional divisor is a subvariety in X isomorphic to P n 1. In particular, X contains a copy of P s P n 1. Now we blow up X along P s to obtain X. The exceptional divisor in X is the projectivization of the normal bundle of P s in X. Since P s is contained in a smooth closed subvariety of dimension r+s+1 in X (choose either P r+s+1 P n 1 if r+s < n 1 or X if r + s = n 1), the normal bundle of P s in X contains a vector subbundle of rank r + 1, and hence its projectivization contains a projective subbundle of rank r. Therefore, X admits a subvariety isomorphic to the total space of a projective bundle of rank r over P s. By (2), the above construction only has an additive effect on the Hodge diamond, i. e. the differences between respective Hodge numbers of X and X are constants independent of X. Hence, we may apply the above construction m 1 more times to obtain a smooth complex projective variety X containing m disjoint copies of the desired projective bundle and satisfying h p,q (X ) h p,q (X) (mod m). In the following, we consider the primitive Hodge numbers l p,q (X) = h p,q (X) h p 1,q 1 (X) for p + q n. Clearly, it suffices to show Theorem 4 for a given collection (l p,q ) (p,q) I of primitive Hodge numbers instead, where I = {(p, q) 1 p q n 1 and p + q n}. This is because one can get back the original Hodge numbers from the primitive Hodge numbers via the relation p h p,q (X) = h 0,q p (X) + l i,q p+i (X) for p q and p + q n, and h 0,q p (X) is a birational invariant. We define a total order on I via (r, s) (p, q) r + s < p + q or (r + s = p + q and s < q). Proposition 6. Let X be a smooth complex projective variety of dimension n. Let (r, s) I. Then X is birational to a smooth complex projective variety X of dimension n such that l r,s (X ) l r,s (X) + 1 (mod m) and for all (p, q) I with (r, s) (p, q). i=1 l p,q (X ) l p,q (X) (mod m) Proof. By Lemma 5, we may assume that X contains m disjoint copies of a projective bundle of rank r 1 over P s r+1. Therefore, it is possible to blow up X along a projective bundle B d of rank r 1 over smooth hypersurfaces Y d P s r+1 of degree d (in case of r = s, Y d just consists of d distinct points in P 1 ) and we may repeat this procedure m times and with different values for d. The Hodge numbers of B d are the same as for the trivial bundle Y d P r 1. By the Lefschetz hyperplane theorem, the Hodge diamond of Y d is the sum of the Hodge diamond of Y 1 = P s r, having nonzero entries only in the middle column, and
7 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER 7 of a Hodge diamond depending on d, having nonzero entries only in the middle row. It is well known (e. g. by computing Euler characteristics as in Section 2) that the two outer entries of this middle row are precisely ( d 1 s r+1). Now we blow up X once along B s r+2 and m 1 times along B 1 and denote the resulting smooth complex projective variety by X. Due to (2) and Künneth s formula, this construction affects the Hodge numbers modulo m in the same way as if we would blow up a single subvariety Z P r 1 X, where Z is a (formal) (s r)dimensional Kähler manifold whose Hodge diamond is concentrated in the middle row and has outer entries equal to ( s r+2 1) s r+1 = 1. In particular, we have h p,q (Z P r 1 ) = 0 unless s r p + q s + r 2 (and p + q has the same parity as s r) and p q s r. On the other hand, h p,q (Z P r 1 ) = 1 if s r p + q s + r 2 and p q = s r. Taking differences in (2), it follows that l p,q (X ) l p,q (X) + h p 1,q 1 (Z P r 1 ) h p n+s 1,q n+s 1 (Z P r 1 ) (mod m) for all p + q n. But we have (p n + s 1) + (q n + s 1) = p + q 2n + 2s 2 2s n 2 s r 2 and hence h p n+s 1,q n+s 1 (Z P r 1 ) = 0 for all (p, q) I by the above remark. Further, l r,s (X ) l r,s (X) + h r 1,s 1 (Z P r 1 ) = l r,s (X) + 1 (mod m) since s r (r 1) + (s 1) s + r 2 and r s = s r. Finally, r + s < p + q implies (p 1) + (q 1) > s + r 2, while r + s = p + q and s < q imply p q > s r, so we have h p 1,q 1 (Z P r 1 ) = 0 in both cases and thus l p,q (X ) l p,q (X) + h p 1,q 1 (Z P r 1 ) = l p,q (X) (mod m) for all (p, q) I with (r, s) (p, q). Proof of Theorem 4. The statement is an immediate consequence of applying Proposition 6 inductively t p,q times to each (p, q) I in the descending order induced by, where t p,q h p,q h p,q (X p,q ) (mod m) and X p,q is the variety obtained in the previous step. 4. Polynomial relations The following principle seems to be classical. Lemma 7. Let N 1 and let S Z N be a subset such that its reduction modulo m is the whole of (Z/mZ) N for infinitely many integers m 2. If f C[x 1,..., x N ] is a polynomial vanishing on S, then f = 0. Proof. Let f C[x 1,..., x N ] be a nonzero polynomial vanishing on S. By choosing a Qbasis of C and a Qlinear projection C Q which sends a nonzero coefficient of f to 1, we see that we may assume that the coefficients of f are rational, hence even integral. Since f 0, there exists a point z Z N such that f(z) 0. Choose an integer m 2 from the assumption which does not divide f(z). Then f(z) 0 (mod m). However, we have z s (mod m) for some s S and thus f(z) f(s) = 0 (mod m), because f Z[x 1,..., x N ]. This is a contradiction.
8 8 MATTHIAS PAULSEN AND STEFAN SCHREIEDER Corollary 8. There are no nontrivial polynomial relations among the Hodge numbers of smooth complex projective varieties of the same dimension. Proof. This follows by applying Lemma 7 to the set S of possible Hodge diamonds, where we consider only a nonredundant quarter of the diamond to take the Hodge symmetries into account. Theorem 2 guarantees that the reductions of S modulo m are surjective even for all integers m 2. In the same way Theorem 2 implies Corollary 8, Theorem 4 yields the following. Corollary 9. There are no nontrivial polynomial relations among the inner Hodge numbers of all smooth complex projective varieties in any given birational equivalence class. Acknowledgements The second author thanks J. Kollár and D. Kotschick for independently making him aware of Kollár s question (answered in Corollary 8 above), which was the starting point of this paper. References [Ara16] D. Arapura, Geometric Hodge structures with prescribed Hodge numbers, Recent Advances in Hodge Theory (M. Kerr and G. Pearlstein, eds.), London Mathematical Society Lecture Note Series, no. 427, Cambridge University Press, 2016, pp [Hun89] B. Hunt, Complex manifold geography in dimension 2 and 3, Journal of Differential Geometry 30 (1989), [KS13] D. Kotschick and S. Schreieder, The Hodge ring of Kähler manifolds, Compositio Mathematica 149 (2013), [Sch15] S. Schreieder, On the construction problem for Hodge numbers, Geometry & Topology 19 (2015), [Sim04] C. Simpson, The construction problem in Kähler geometry, Different Faces of Geometry (S. Donaldson, Y. Eliashberg, and M. Gromov, eds.), International Mathematical Series, vol. 3, Springer, 2004, pp [Voi03] C. Voisin, Hodge theory and complex algebraic geometry I, Cambridge studies in advanced mathematics, no. 76, Cambridge University Press, Mathematisches Institut, LudwigMaximiliansUniversität München, Theresienstr. 39, D München, Germany address: Mathematisches Institut, LudwigMaximiliansUniversität München, Theresienstr. 39, D München, Germany address:
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