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Characteristic numbers of algebraic varieties

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 1, 2009 (received for review March 30, 2009)
Abstract
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least 3, only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation preserving. In the space of Chern numbers, there are 2 distinguished subspaces, one spanned by the Euler and Pontryagin numbers, and the other spanned by the Hirzebruch–Todd numbers. Their intersection is the span of the Euler number and the signature.
In 1954, Hirzebruch (problem 31 in ref. 1) asked which linear combinations of Chern numbers of smooth complex projective varieties are topologically invariant. The purpose of this article is to announce a complete solution to this problem and to discuss some additional results about the characteristic numbers of algebraic varieties that arise from this solution. Some of these further results characterize the characteristic numbers of smooth complex projective varieties that can be bounded in terms of Betti numbers. Another one of these results, Theorem 6 below, determines the intersection of Pontryagin and Hirzebruch–Todd numbers in even complex dimensions.
Solution of Hirzebruch's Problem
Because the manifold underlying a complexalgebraic variety has a preferred orientation, it is most natural to interpret the term “topological invariance” in Hirzebruch's problem to mean invariance under orientationpreserving homeo or diffeomorphisms. With this interpretation the solution to the problem is given by:
Theorem 1.
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers.
In one direction, the Euler number, which is the top Chern number, is of course a homotopy invariant. Further, the Pontryagin numbers, which are special linear combinations of Chern numbers, are oriented diffeomorphism invariants. In fact, Novikov proved that the Pontryagin numbers are also invariant under orientationpreserving homeomorphisms, and so Theorem 1 is unchanged if we replace oriented diffeomorphism invariance by oriented homeomorphism invariance. The other direction, proving that there are no other linear combinations that are oriented diffeomorphism invariants, has proved to be quite difficult because of the scarcity of examples of diffeomorphic algebraic varieties with distinct Chern numbers.
Given Theorem 1, and the fact that Pontryagin numbers depend on the orientation, one might expect that only the Euler number is invariant under homeo or diffeomorphisms that do not necessarily preserve the orientation. For diffeomorphisms, this is almost but not quite true:
Theorem 2.
In complex dimension n ≥ 3, a rational linear combination of Chern numbers is a diffeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler number c_{n}. In complex dimension 2, both Chern numbers c_{2} and c_{1}^{2} are diffeomorphism invariants of complex surfaces.
The statement about complex dimension 2 is a consequence of Seiberg–Witten theory. It is an exception due to the special nature of 4dimensional differential topology. The exception disappears if we consider homeomorphisms instead of diffeomorphisms:
Theorem 3.
A rational linear combination of Chern numbers is a homeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler number.
These theorems show that linear combinations of Chern numbers of complex projective varieties are not usually determined by the underlying manifold. This motivates the investigation of a modification of Hirzebruch's original problem, asking how far this indeterminacy goes. More precisely, one would like to know which linear combinations of Chern numbers are determined up to finite ambiguity by the topology. The obvious examples for which this is true are the numbers By the Hirzebruch–Riemann–Roch theorem (see ref. 2) these are indeed linear combinations of Chern numbers. By their very definition, together with the Hodge decomposition of the cohomology, the numbers χ_{p} are bounded above and below by linear combinations of Betti numbers. It turns out that this property characterizes the linear combinations of the χ_{p}, as shown by the following theorem:
Theorem 4.
A rational linear combination of Chern numbers of complex projective varieties can be bounded in terms of Betti numbers if and only if it is a linear combination of the χ_{p}.
The span of the χ_{p} includes the Euler number c_{n} = Σ_{p} (−1)^{p}χ_{p} and the signature, which, according to the Hodge index theorem, equals Σ_{p}χ_{p}. It also includes the Chern number c_{1}c_{n−1}, by a result of Libgober and Wood (theorem 3 in ref. 3). Nevertheless, the span of the χ_{p} is a very small subspace of the space of linear combinations of Chern numbers. The latter has dimension equal to π(n), the number of partitions of n, which grows exponentially with n. The former has dimension [(n + 2)/2], the integral part of (n + 2)/2. This follows from the symmetries of the Hodge decomposition, which imply χ_{p} = (−1)^{n}χ_{n−p}, together with the linear independence of χ_{0}, … , χ_{[n/2]}, which can be checked, for example, by evaluating on all ndimensional products of ℂP^{1} and ℂP^{2}.
The proofs of Theorems 1, 2, 3, and 4 are carried out in ref. 4.
Euler–Pontryagin versus Hirzebruch–Todd
We now restrict ourselves to even complex dimensions n = 2m. Consider the ℚvector space C_{2m} of linear combinations of Chern numbers of almost complex manifolds of complex dimension 2m, equivalently of real dimension 4m. This is a vector space of dimension π(2m), the number of partitions of 2m, with a basis given by the monomials c_{i1}· … ·c_{ik} with Σ_{j}i_{j} = 2m.
Define ℰ𝒫_{2m} ⊂ C_{2m} to be the subspace spanned by the Euler number e = c_{2m} and by the Pontryagin numbers p_{i1}· … ·p_{ik} with Σ_{j}i_{j} = m, where are the Pontryagin classes. This subspace has dimension 1 + π(m).
In the same spirit as Theorem 4 we have the following:
Theorem 5.
A rational linear combination of Pontryagin numbers of complex projective varieties can be bounded in terms of Betti numbers if and only if it is a multiple of the signature.
In other words, an element of ℰ𝒫_{2m} evaluated on projective varieties can be bounded in terms of Betti numbers if and only if it is a linear combination of the Euler number and the signature. Theorem 5 is implicit in refs. 4 and 5 but is not explicitly stated there. Corollary 3 of ref. 5 says that a rational linear combination of Pontryagin numbers of smooth manifolds can be bounded in terms of Betti numbers if and only if it is a multiple of the signature. The proof given in ref. 5 uses sequences of ring generators for the oriented bordism ring Ω_{★} ⊗ ℚ, which, in dimensions ≥8, are ℂP^{2k}bundles over S^{4}, and are not complexalgebraic varieties. However, in that proof, one can replace S^{4} by T^{4} endowed with the algebraic structure of an Abelian surface, and one can assume that the ℂP^{2k}bundles over it are algebraic, so as to obtain a proof of Theorem 5 above. This amounts to mapping the proof of Theorem 4 given in ref. 4 from Ω_{★}^{U}⊗ℚ to Ω_{★}⊗ℚ under the forgetful natural transformation.
Next, define ℋ𝒯_{2m} ⊂ C_{2m} to be the subspace spanned by the Hirzebruch–Todd numbers T_{2m}^{p} defined in section 1.8 of ref. 2. These are certain universal linear combinations of Chern numbers, which occur in the expansion where T_{2m}(y; c_{1}, … , c_{2m}) is the multiplicative sequence corresponding to the power series It follows from the definition that T_{2m}^{p} = T_{2m}^{2m−p}, so that there are at most m + 1 distinct T_{2m}^{p}, namely T_{2m}^{0}, … , T_{2m}^{m}. By evaluating on products of projective spaces, one sees that these m + 1 linear combinations of Chern numbers are indeed linearly independent over ℚ, so that ℋ𝒯_{2m} is of dimension m + 1.
We have the following result about the relation between Pontryagin and Hirzebruch–Todd numbers:
Theorem 6.
The intersection ℰ𝒫_{2m} ∩ ℋ𝒯_{2m} is 2dimensional, spanned by the Euler number and the signature.
The statement is vacuous for m = 1, as C_{2} = ℰ𝒫_{2} = ℋ𝒯_{2}. In all dimensions, it is clear that the Euler number c_{2m} = Σ_{p}(−1)^{p}T_{2m}^{p} is in the intersection. The signature is a linear combination of Pontryagin numbers by a result of Thom, and so is in ℰ𝒫_{2m}. The space ℋ𝒯_{2m} of Hirzebruch–Todd numbers contains the Lgenus L_{m} (see section 1.8 of ref. 2). By the Hirzebruch signature theorem, L_{m} equals the signature. So to prove Theorem 6, we have to prove that the only Pontryagin numbers in ℰ𝒫_{2m} ∩ ℋ𝒯_{2m} are the multiples of the signature.
First proof of Theorem 6.
The Hirzebruch–Riemann–Roch theorem implies that in complex dimension 2m the χ_{p}numbers in Eq. 1 are equal to the Hirzebruch–Todd numbers T_{2m}^{p}. In the special case when we evaluate on a smooth complex projective variety, or, more generally, on a compact Kähler manifold, the Hodge numbers h^{p,q} are bounded by the Betti numbers because of the Hodge decomposition of cohomology. Therefore, in this case, the Hirzebruch–Todd numbers T_{2m}^{p} are bounded by linear combinations of Betti numbers, and Theorem 6 follows from Theorem 5. This completes our first proof of Theorem 6.
One can also give a more direct proof of Theorem 6, without appealing to Theorem 5.
Second proof of Theorem 6.
The rational complex cobordism ring Ω_{★}^{U}⊗ℚ is a polynomial ring on generators γ_{1},γ_{2},…. The proof of Theorem 4 in ref. 4 provides a specific choice for the γ_{i}, which, for i ≥ 3, are in the kernel of the Hirzebruch χ_{y}genus defined by χ_{y} = Σ_{p}χ_{p}y^{p}. In fact, because products of ℂP^{1} and ℂP^{2} detect all of the components of the χ_{y}genus, any basis sequence can be modified by the addition of such products to achieve γ_{i} ∈ kerχ_{y} for i ≥ 3.
The Hirzebruch–Riemann–Roch theorem (2) implies that the χ_{p}numbers agree with the Hirzebruch–Todd numbers T_{2m}^{p}, so that the subspace ker_{χy} ⊂ Ω_{2m}^{U}⊗ℚ, which is spanned by monomials containing a γ_{i} with i ≥ 3, coincides with the annihilator of ℋ𝒯_{2m}.
The annihilator of the Pontryagin numbers is the kernel of the forgetful map Ω_{★}^{U}⊗ℚ → Ω_{★}⊗ℚ, which is the ideal generated by the γ_{i} with odd i.
Now, if f ∈ ℰ𝒫_{2m} ∩ ℋ𝒯_{2m}, then its kernel in the bordism group Ω_{2m}^{U}⊗ℚ contains all monomials containing a γ_{i} with i ≥ 3, and it also contains all monomials containing a γ_{i} with i odd. Therefore, the only monomial in the generators not contained in ker f is γ_{2}^{m}. If we normalize f then it follows that f = L_{m}.
This completes our second proof of Theorem 6.
It would be interesting to know whether there is an elementary algebraic or combinatorial proof of Theorem 6, showing directly from the definition (Eq. 2) of the Hirzebruch–Todd numbers that their span contains no Pontryagin numbers other than the multiples of L_{m}, without using bordism theory or the Hirzebruch–Riemann–Roch theorem.
Acknowledgments
I gratefully acknowledge several helpful comments and suggestions from P. Landweber and the support of The Bell Companies Fellowship at the Institute for Advanced Study in Princeton.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: dieter{at}member.ams.org

Author contributions: D.K. performed research and wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.
References
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 Hirzebruch F
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 Libgober AS,
 Wood JW
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 Kotschick D
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