Chern numbers and diffeomorphism types of projective varieties
aa r X i v : . [ m a t h . G T ] D ec CHERN NUMBERS AND DIFFEOMORPHISM TYPES OF PROJECTIVE VARIETIES
D. KOTSCHICK
Herrn Prof. Dr. F. Hirzebruch zum 80. Geburtstag gewidmet. A BSTRACT . In 1954 Hirzebruch asked which linear combinations of Chern numbers are topologicalinvariants of smooth complex projective varieties. We give a complete answer to this question insmall dimensions, and also prove partial results without restrictions on the dimension.
1. I
NTRODUCTION
More than fifty years ago, Hirzebruch raised the question to what extent the Chern and Hodgenumbers of projective algebraic manifolds are topologically invariant, see Problem 31 in [2]. Henoted that Chern numbers of almost complex manifolds are not topologically invariant simplybecause there are too many almost complex structures, even on a fixed manifold. Then, in 1959,Borel and Hirzebruch gave an example of a -dimensional closed oriented manifold with twoprojective algebraic structures for which c are different, see [1] Section 24.11. Until 1987, whenthe commentary in [4] was written, nothing further was discovered concerning this question. Inparticular, Hirzebruch wrote then that he did not know whether c and c are topological invariantsof complex three- and four-folds respectively.In this paper we prove that in complex dimension the only linear combinations of the Chernnumbers c , c c and c that are invariant under orientation-preserving diffeomorphims of simplyconnected projective algebraic manifolds are the multiples of the Euler characteristic c . In dimen-sion the only linear combinations of Chern numbers that are invariant are the linear combinationsof the Euler characteristic c and of the Pontryagin numbers p and p . We also prove some partialresults in arbitrary dimensions.These results stem from the fact that in complex dimension the Chern number c is not invariantunder orientation-reversing homeomorphisms; cf. [5]. By suitable stabilisation of the counterex-amples, we find enough examples at least in dimensions and to detect the independent variationof all Chern numbers which are not combinations of the Euler and Pontryagin numbers. Our re-sults suggest a weaker form of Hirzebruch’s problem, asking whether the topology determines theChern numbers up to finite ambiguity. We have no counterexample to an affirmative answer to thisweaker question in the projective algebraic case, although it is known to be false for non-K¨ahlercomplex manifolds; cf. [7]. 2. P RELIMINARY RESULTS
Complex surfaces.
For complex surfaces there are two Chern numbers, c and c , which turnout to be diffeomorphism invariants even without assuming that the diffeomorphism is orientation-preserving with respect to the orientations given by the complex structures: Date : December 20, 2007; MSC 2000: primary 57R20, secondary 14J30, 14J35, 32Q55.
Theorem 1.
If two compact complex surfaces are diffeomorphic, then their Chern numbers coin-cide.Proof.
In this case c is the topological Euler characteristic e . By Wu’s formula we have(1) c ( X ) = 2 e ( X ) + p ( X ) . The first Pontryagin number is times the signature, and so the right-hand side of (1) is invari-ant not just under orientation-preserving diffeomorphisms, but even under orientation-preservinghomotopy equivalences .Now suppose that two compact complex surfaces are orientation-reversing diffeomorphic, withrespect to the orientations defined by their complex structures. Then, using Seiberg–Witten theory,I proved in 1995, see Theorem 2 of [6], that the signatures of these surfaces vanish. Thus, theirChern numbers agree by (1). (cid:3) The statement about orientation-reversing diffeomorphisms concerns projective algebraic sur-faces only, because, by the classification of complex surfaces, a complex surface with positivesignature is always projective.As we saw in the proof, c is invariant under orientation-preserving homotopy equivalences, andunder orientation-reversing diffeomorphisms. But it is not invariant under orientation-reversinghomeomorphisms. Already in 1991, I had proved the following: Theorem 2 ([5]) . There are infinitely many pairs of simply connected projective algebraic surfaces X i and Y i of non-zero signature which are orientation-reversing homeomorphic. The proof is based on geography results for surfaces of general type due to Persson and Chen.The surfaces X i and Y i are projective algebraic because they are of general type. They can be cho-sen to contain embedded holomorphic spheres, in which case they can not be orientation-reversingdiffeomorphic, although they are orientation-reversing homeomorphic. This was the motivationfor the results of [6] quoted in the proof of Theorem 1.By Wu’s formula (1) the homeomorphic surfaces X i and Y i have different c . Indeed, the homeo-morphism in question preserves the Euler number and reverses the sign of the signature, so that (1)gives:(2) c ( Y i ) = 4 e ( X i ) − c ( X i ) . Wu’s formula (1) shows in particular that the unoriented homeomorphism type almost determinesthe Chern numbers of a compact complex surface: there are only two possible values for c (andonly one for c , of course).We shall use the examples from Theorem 2 as building blocks for our high-dimensional exam-ples.2.2. Inductive formulae for Chern classes.
We require the following easy calculation.
Lemma 1.
Let A be a compact complex n -fold, and B = A × C P . Then the Chern numbers of B are (3) c r . . . c r k ( B ) = 2 k X j =1 c r . . . c r j − . . . c r k ( A ) . All this was known in 1954, and Hirzebruch [2] remarked that the Chern numbers of an algebraic surface aretopological invariants (of the underlying oriented manifold).
HERN NUMBERS AND DIFFEOMORPHISM TYPES 3
Proof.
The Whitney sum formula c ( T B ) = c ( T A ) c ( T C P ) for the total Chern classes impliesthat, with respect to the K¨unneth decomposition of the cohomology of B , the Chern classes of B are c ( B ) = c ( A ) + c ( C P ) c ( B ) = c ( A ) + c ( A ) c ( C P ) . . .c n ( B ) = c n ( A ) + c n − ( A ) c ( C P ) c n +1 ( B ) = c n ( A ) c ( C P ) . The claim follows using that the first Chern number of C P equals . (cid:3) We also need the following generalization of Lemma 1 to non-trivial C P -bundles: Lemma 2.
Let B be a compact complex surface and E −→ B a holomorphic vector bundle ofrank two. Then the projectivisation X = P ( E ) has c ( X ) = 2 c ( B ) ,c c ( X ) = 2( c ( B ) + c ( B )) ,c ( X ) = 6 c ( B ) + 2 p ( P ( E )) . (4)Here p ( P ( E )) = c ( E ) − c ( E ) is the first Pontryagin number for the group SO (3) = P U (2) ,which is the structure group of the sphere bundle X −→ B . Notice that in the case that p ( P ( E )) =0 , the formulae reduce to those obtained for the trivial bundle. Proof.
The formulae for c and for c c are immediate from the multiplicativity of the topologicalEuler characteristic and of the Todd genus, recalling that the Todd genera in dimension and are ( c + c ) , respectively c c . To compute c note that by the Leray-Hirsch theorem thecohomology ring of X is generated as a H ∗ ( B ) -module by a class y ∈ H ( X ) restricting as agenerator to every fiber and satisfying the relation y + c ( E ) y + c ( E ) = 0 . Moreover, c ( X ) = c ( B ) + c ( E ) + 2 y because the vertical tangent bundle has first Chern class c ( E ) + 2 y . The third power is computed straightforwardly using the relation and the fact that y evaluates to on the fiber. (cid:3)
3. C
OMPLEX THREE - FOLDS
A variant of Hirzebruch’s problem for three-folds was taken up by LeBrun in 1998, see [7],who proved that there are closed -manifolds which admit complex structures with different c c and c . He even proved that a fixed manifold can have complex structures realising infinitelymany different values for c c . However, for all the examples discussed in [7] only one of thecomplex structures is projective algebraic, or at least K¨ahler, and all the others are non-K¨ahler.Therefore, these examples say nothing about the topological invariance of Chern numbers forprojective algebraic three-folds.Nevertheless, both c c and c are not diffeomorphism invariants of projective three-folds: Proposition 1.
There are infinitely many pairs of projective algebraic three-folds Z i and T i withthe following properties: D. KOTSCHICK (i)
Each Z i and T i admits an orientation-reversing diffeomorphism. (ii) For each i the manifolds underlying Z i and T i are diffeomorphic. (iii) For each i one has c c ( Z i ) = c c ( T i ) and c ( Z i ) = c ( T i ) .Proof. Let X i and Y i be the algebraic surfaces from Theorem 2, constructed in [5], and take Z i = X i × C P and T i = Y i × C P . Then the identity on the first factor times complex conjugation onthe second factor gives an orientation-reversing selfdiffeomorphism of Z i and of T i .Denote by ¯ Y i the smooth manifold underlying Y i , but endowed with the orientation oppositeto the one induced by the complex structure. Then X i and ¯ Y i are orientation-preserving home-omorphic simply connected smooth four-manifolds, and are therefore h-cobordant. If W is anh-cobordism between them, then W × S is an h-cobordism between Z i and ¯ T i = ¯ Y i × C P .By Smale’s h-cobordism theorem, Z i and ¯ T i are orientation-preserving diffeomorphic. As Z i and T i admit orientation-reversing diffeomorphisms, we conclude that they are both orientation-preserving and orientation-reversing diffeomorphic.For the Chern numbers (3) gives c c ( Z i ) = 2( c + c )( X i ) ,c ( Z i ) = 6 c ( X i ) , (5)and similarly for T i and Y i . As X i and Y i have the same c but different c , we conclude that Z i and T i have different c c and different c . (cid:3) Thus c c and c are not topological invariants of projective three-folds, but it is not yet clearthat they vary independently. This is the content of the following: Theorem 3.
The only linear combinations of the Chern numbers c , c c and c that are invariantunder orientation-preserving diffeomorphisms of simply connected projective algebraic three-foldsare the multiples of the Euler characteristic c .Proof. First of all, let us dispose of the orientation question. If two complex three-folds areorientation-reversing diffeomorphic with respect to the orientations given by their complex struc-tures, then they become orientation-preserving diffeomorphic after we replace one of the complexstructures by its complex conjugate. As the conjugate complex structure has the same Chernnumbers as the original one, we do not have to distinguish between orientation-preserving andorientation-reversing diffeomorphisms.All the examples constructed in the proof of Proposition 1 have the property that c c − c agrees on Z i and T i , as follows by combining (2) with (5) and the topological invariance of c forsurfaces. But, by Proposition 1, linear combinations of c c − c and of c are the only candidatesleft for combinations of Chern numbers that can be topological invariants of projective three-folds.In order to show that c c − c is not an oriented diffeomorphism invariant we shall use certainruled manifolds which are non-trivial C P -bundles, rather than the products used above.Consider again a pair X i and Y i of simply connected algebraic surfaces as in Theorem 2. Forsimplicity we just denote them by X and Y , with orientations implicitly given by the complexstructures. The oriented manifolds X and ¯ Y are orientation-preserving h-cobordant. Let M be theprojectivisation of the holomorphic tangent bundle T Y of Y . Temporarily ignoring the complexstructure of M , we think of it as a smooth oriented two-sphere bundle over Y , or over ¯ Y . If W isany h-cobordism between X and ¯ Y , then the two-sphere bundle M −→ ¯ Y extends to a uniquelydefined oriented two-sphere bundle V −→ W . Let N be the restriction of this bundle to X ⊂ W .If we give N the orientation induced from that of X and M the orientation induced from that of ¯ Y , HERN NUMBERS AND DIFFEOMORPHISM TYPES 5 then, by construction, V is an h-cobordism between N and M . By Smale’s h-cobordism theorem, M and N are diffeomorphic.Because the bundle p : M −→ Y was defined as the projectivisation of the holomorphic tangentbundle of Y , its characteristic classes are w ( p ) = w ( Y ) and p ( p ) = c ( Y ) − c ( Y ) . Consideredas a bundle over ¯ Y , p has the same Stiefel–Whitney class, but the first Pontryagin number changessign. It follows that q : N −→ X has p ( q ) = − c ( Y ) + 4 c ( Y ) = c ( X ) , where the last equality is from (2). Moreover, w ( q ) = w ( X ) , although X and ¯ Y are not dif-feomorphic. This follows for example from the cohomological characterisation of w ( X ) as theunique element of H ( X ; Z ) which for all x satisfies w ( X ) · x ≡ x (mod 2) . The bundle q is determined by w ( q ) = w ( X ) and p ( q ) = c ( X ) , and so we can think of itas the projectivisation of the holomorphic rank two bundle O ( K ) ⊕ O −→ X . Therefore the totalspace N inherits a complex-algebraic structure from that of X . Its Chern numbers are given by (4): c ( N ) = 2 c ( X ) ,c c ( N ) = 2( c ( X ) + c ( X )) ,c ( N ) = 8 c ( X ) . (6)This N is diffeomorphic to M , which has a complex-algebraic structure as the projectivisation ofthe holomorphic tangent bundle of Y . (Recall from the beginning of the proof that we do not haveto keep track of the orientations induced by complex structures, because we can always replace astructure by its complex conjugate.) The Chern numbers of M are also given by (4): c ( M ) = 2 c ( Y ) = 2 c ( X ) ,c c ( M ) = 2( c ( Y ) + c ( Y )) = 2( − c ( X ) + 5 c ( X )) ,c ( M ) = 8 c ( Y ) − c ( Y ) = 8( − c ( X ) + 3 c ( X )) , (7)using (2) to replace the Chern numbers of Y by combinations of those of X . Unlike for theexamples in Proposition 1, the combination c c − c is not the same for M and N . This finallyshows that c c and c vary independently (within a fixed diffeomorphism type). (cid:3) Although the Chern numbers of a projective three-fold are not determined by the underlyingdifferentiable manifold, this may still be the case up to finite ambiguity. By the Hirzebruch–Riemann–Roch theorem one has(8) c c = 1 − h , + h , − h , , so that in the K¨ahler case c c is bounded from above and from below by linear combinations ofBetti numbers. In particular, for K¨ahler structures on a fixed -manifold c c can take at mostfinitely many values. We are left with the following: Problem 1.
Does c take on only finitely many values on the projective algebraic structures withthe same underlying -manifold? The issue here is that there is no Riemann–Roch type formula expressing c as a combination ofHodge numbers and the other Chern numbers. D. KOTSCHICK
For three-folds with ample canonical bundle one has c < , and Yau’s celebrated work [11]gives c ≥ c c . As c c is bounded below by a linear combination of Betti numbers, we havea positive answer to Problem 1 for this restricted class of projective three-folds. Even in the non-K¨ahler category, there are no examples where infinitely many values are known to arise for c .4. H IGHER DIMENSIONS
It is now very easy to show that, except for the Euler number, no Chern number is diffeomorphism-invariant:
Theorem 4.
For projective algebraic n -folds with n ≥ the only Chern number c I which isdiffeomorphism-invariant is the Euler number c n . Note that by Theorem 1 this is false for n = 2 , because in that case c is also diffeomorphism-invariant. On the other hand, by Theorem 2 it is not homeomorphism-invariant, so that Theorem 4is true for n = 2 if we replace diffeomorphism-invariance by homeomorphism-invariance. As inthe case of Proposition 1, the examples we exhibit in the proof of Theorem 4 admit orientation-reversing diffeomorphisms, so that one cannot restore diffeomorphism-invariance of c I = c n byrestricting to orientation-preserving diffeomorphisms only. Proof.
For n = 3 this was already proved in Proposition 1. For n > we take the examples T i and Z i from Proposition 1 and multiply them by n − copies of C P . Call these products T ′ i and Z ′ i .Using formula (3) and induction, we see that, on the one hand, c n is a universal multiple of the c ofthe surfaces we started with. On the other hand, c n ( T ′ i ) and c n ( Z ′ i ) are universal multiples of c ( X i ) and of c ( Y i ) respectively, and so are different. All other Chern numbers c I are universal linearcombinations of c ( X i ) and c ( X i ) , respectively c ( Y i ) and c ( Y i ) , with the coefficients of both c and c strictly positive. As X i and Y i have the same c but different c , the result follows. (cid:3) Although the individual Chern numbers are not diffeomorphism-invariant, certain linear com-binations are invariant once we restrict to orientation-preserving diffeomorphisms. Of course, asremarked by Hirzebruch [4], the Pontryagin numbers p J have this invariance property , but thisonly helps when the complex dimension is even. Problem 2.
Prove that, in arbitrary dimensions, the only combinations of Chern numbers that areinvariant under orientation-preserving diffeomorphisms of smooth complex projective varieties arelinear combinations of Euler and Pontryagin numbers.
For complex dimension this is Theorem 3 above. Theorem 5 below deals with the case ofcomplex dimension .It would be interesting to know whether each of the Chern numbers c I = c n takes on only finitelymany values on a fixed smooth manifold. In the K¨ahler case this is known to be true for c c n − bya result of Libgober and Wood, who showed that this Chern number is always a linear combinationof Hodge numbers, see Theorem 3 in [8]. In the non-K¨ahler case c c n − can take on infinitelymany values on a fixed manifold. This follows as in the proof of Theorem 4 by taking productsof LeBrun’s examples [7] mentioned in the previous section with C P , and using formula (3).Because c c takes on infinitely many values on a fixed -manifold, the same conclusion holds for c c n − in real dimension n ≥ . Note that, unlike the Euler number, the Pontryagin numbers change sign under a change of orientation.
HERN NUMBERS AND DIFFEOMORPHISM TYPES 7
Returning to the K¨ahler case, the Riemann–Roch theorem expresses the χ p -genus χ p = n X q =0 ( − q h p,q as a linear combination of Chern numbers, and it follows that the combinations of Chern numberswhich appear in this way can take on only finitely many values on a fixed manifold, as they arebounded above and below by linear combinations of Betti numbers. Remark . If the complex dimension n is odd, then the Todd genus expressing the Euler character-istic χ = ( − n χ n of the structure sheaf as a combination of Chern numbers does not involve c n .This follows from the Bemerkungen in Section 1.7 of [3]. On the one hand, in any dimension, thecoefficient of c n in the Todd genus agrees with the coefficient of c n . On the other hand, the Toddgenus is divisible by c if n is odd.Generalising this Remark, and what we saw for n = 3 in the previous section, we now prove: Proposition 2. If M is K¨ahler of odd complex dimension n > , then all χ p are linear combina-tions of Chern numbers which do not involve c n . This shows that for odd n there is no general way to extract the value of c n from the Hodgenumbers. In particular, one can not obtain a finiteness result for c n in this way. Proof.
The K¨ahler symmetries imply χ p = ( − n χ n − p , so that it is enough to prove the claim for p > n . We shall do this by descending induction starting at p = n .Salamon [10] proved that for ≤ k ≤ n the number(9) n X p = k ( − p (cid:18) pk (cid:19) χ p is a linear combination of Chern numbers each of which involves a c i with i > n − k ] , see [10]Corollary 3.3. Using this for n odd and k = n , we obtain once more the claim for χ n treatedalready in the Remark above.Suppose now that the claim has been proved for χ n , χ n − , . . . , χ j with j > n + 1 . Then weconsider (9) with k = j − . (Note that this still satisfies k ≥ .) As χ p with p ≥ j does not involve c n by the induction hypothesis, Salamon’s result implies that χ j − does not involve c n either. (cid:3)
5. F
OUR - FOLDS
In the case of four-folds, in addition to the Euler number c , the following are invariants of theunderlying oriented smooth manifold: p = ( c − c ) = c − c c + 4 c p = c − c c + 2 c . (10)The vector space of Chern numbers of four-folds is -dimensional, containing the -dimensionalsubspace spanned by c , p and p . It turns out that all combinations of Chern numbers that areinvariant under orientation-preserving diffeomorphisms are contained in this subspace: Our notation is consistent with [10], changing the traditional superscript in χ p from [3] to a subscript. D. KOTSCHICK
Theorem 5.
The only linear combinations of Chern numbers that are invariant under orientation-preserving diffeomorphisms of simply connected projective algebraic four-folds are linear combi-nations of the Euler characteristic and of the Pontryagin numbers.Proof.
This is a rather formal consequence of our results for complex three-folds. Consider thevector space of Chern number triples ( c , c c , c ) . Whenever we have a smooth six-manifold withtwo different complex structures, the difference of the two Chern vectors must be in the kernel ofany linear functional corresponding to a topologically invariant combination of Chern numbers.In the proof of Theorem 3 we produced two kinds of examples for which these difference vectorswere linearly independent. Therefore the space of topologically invariant combinations of Chernnumbers is at most one-dimensional, and as it contains c it is precisely one-dimensional.Consider now the four-folds obtained by multiplying the three-dimensional examples by C P .If the difference of Chern vectors in a three-dimensional example is (0 , a, b ) , then by (3) the dif-ference of Chern vectors ( c , c c , c , c c , c ) for the product with C P is (0 , a, a, a + 2 b, b ) .Two examples in dimension three with linearly independent difference vectors lead to examplesin dimension four which also have linearly independent difference vectors. Thus, in the five-dimensional space spanned by the Chern numbers of complex projective four-folds, the subspaceinvariant under orientation-preserving diffeomorphisms has codimension at least two. As it con-tains the linearly independent elements c , p and p , it is exactly three-dimensional. (cid:3) Concerning the weaker question which Chern numbers of projective or K¨ahler four-folds aredetermined by the topology up to finite ambiguity, this is so for c c on general grounds, see thediscussion above and [8, 10]. The formula for p then shows that c is also determined up to finiteambiguity. Using either the formula for p or the Riemann–Roch formula for the structure sheaf,we conclude that c − c c is also determined up to finite ambiguity, but it is not clear whetherthis is true for c and c c individually. Note that a negative answer to Problem 1, giving infinitelymany values for c on a fixed -manifold, would show that c also takes on infinitely many valueson a fixed -manifold by taking products with C P .For four-folds with ample canonical bundle one has c > , and Yau’s work [11] gives c ≤ c c . Therefore < c ≤
53 (4 c c − c ) . As the right-hand side takes on only finitely many values, the same is true for c , and then for c c as well. Remark . Pasquotto [9] recently raised the question of the topological invariance of Chern num-bers of symplectic manifolds, particularly in (real) dimensions and . Our results for K¨ahlermanifolds of course show that Chern numbers of symplectic manifolds are not topological invari-ants. In the K¨ahler case we have used Hodge theory to argue that the variation of Chern numbersis quite restricted, often to finitely many possibilities. It would be interesting to know whether anyfiniteness results hold in the symplectic non-K¨ahler category.R EFERENCES
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UDWIG -M AXIMILIANS -U NIVERSIT ¨ AT M ¨
UNCHEN , T
HERESIENSTR . 39, 80333M ¨
UNCHEN , G
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