EINSTEIN METRICS AND SMOOTH STRUCTURES

D. KOTSCHICK

Abstract. We prove that there are infinitely many pairs of homeomorphic non–diffeomorphic smooth 4–manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. We also show that there are closed 4–manifolds with two smooth structures which admit Einstein metrics with opposite signs of the scalar curvature.

In dimensions strictly smaller than four Einstein metrics have constant curvature and are therefore very rare. In dimension four Einstein metrics of non–constant curvature exist, but it is still the case that existence of such a metric imposes non–trivial restrictions on the underlying manifold^{1}. For closed orientable Einstein 4–manifolds X the Euler characteristic has to be non–negative, and, furthermore, the Hitchin–Thorpe inequality

e(X) ≥ \frac{3}{2}|σ(X)| (1)

must hold, where e denotes the Euler characteristic and σ the signature. This condition is very crude, and is certainly homotopy invariant, as are the restrictions coming from Gromov’s notion of simplicial volume [12], and from the existence of maps of non–zero degree to hyperbolic manifolds [4].

Our aim in this note is to discuss existence and non–existence of Einstein metrics as a property of the smooth structure. We shall exhibit infinitely many pairs of homeomorphic non–diffeomorphic smooth 4–manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. Such examples have not appeared before, although in the context of Seiberg–Witten theory they are not unexpected, and are related to a recent paper of LeBrun [19].

An isolated example of such a pair can be obtained as follows. Hitchin [13] showed that Einstein manifolds for which (1) is an equality are either flat or quotients of a K3 surface with a Calabi–Yau metric. Thus, the existence of smooth manifolds homeomorphic but not diffeomorphic to the K3 surface, which is known from Donaldson theory [11] and also follows easily from Seiberg–Witten theory, see for example [16], shows that by changing only the differentiable structure one can pass from a manifold with an Einstein metric to one without. The point of our examples is that there are lots of them, and they do not arise from the borderline case of a non–existence result. They are in some sense generic.

1991 Mathematics Subject Classification. 57R55,57R57,53C25,14J29.

The author is grateful to the Department of Mathematics at Brown University and to the Max–Planck–Institute for hospitality and support.

1No such restrictions are known in higher dimensions.

We shall also discuss a conjecture concerning uniqueness of Einstein metrics on 4–manifolds which complements the discussion of existence. This too depends on a consideration of different smooth structures on a fixed topological manifold.

1. Smooth structures as obstructions

We shall use Seiberg–Witten invariants to show that certain smooth structures obstruct the existence of Einstein metrics, and refer the reader to [23, 8, 16] for the definitions and basic properties of the invariants. All manifolds in this section are closed, smooth, oriented 4–manifolds. For the sake of simplicity, we assume b_{2}^{+} > 1 throughout, though this is not essential.

We shall need the following result concerning the behaviour of the invariants under connected summing with \overline{ℂP^{2}}. This is usually referred to as a blowup formula.

Proposition 1 ([9, 17]). Let P̃(Y) be a Spin^{c}–structure on Y, and X = Y # \overline{ℂP}^{2}, with E a generator of H^{2}(\overline{ℂP}^{2}, ℤ). Then X has a Spin^{c}–structure P̃(X) with c_{1}(P̃(X)) = c_{1}(P̃(Y)) + E, such that the Seiberg–Witten invariants of P̃(Y) and of P̃(X) are equal (up to sign).

As the reflection in E^{⊥} in the cohomology of X is realised by a selfdiffeomorphism, the naturality of the invariants shows that there is another Spin^{c}–structure with the same Seiberg–Witten invariant, up to sign, and with c_{1}(P̃(X)) = c_{1}(P̃(Y)) − E.

Using this, we can prove the following generalisation of a theorem in [19]:

Theorem 2. Let Y be a manifold with a non–zero Seiberg–Witten invariant, and X = Y #k \overline{ℂP}^{2}. If k > \frac{2}{3}(2e(Y) + 3σ(Y)), then X does not admit an Einstein metric.

Proof. If P̃(X) has a non–zero Seiberg–Witten invariant, then for every Riemannian metric g there must be a solution (A, ϕ) of the monopole equations. Denoting by Â the connection induced by the Spin^{c}– connection A on the determinant bundle of the spinor bundle, we have

c_{1}^{2}(P̃(X)) = \frac{1}{4π^{2}} ∫_{X}(|F + ; Â|^{2} − |F − ; Â|^{2})dvol_{g} ≤ \frac{1}{4π^{2}} ∫_{X} |F + ; Â|^{2}dvol_{g}

= \frac{1}{32π^{2}} ∫_{X} |ϕ|^{4}dvol_{g} ≤ \frac{1}{32π^{2}} ∫_{X} s_{g}^{2}dvol_{g} ,

where s_{g} denotes the scalar curvature of g.

Given any class c ∈ H^{2}(X, ℝ), denote by c^{+} the projection of c into the subspace H_{+}^{2} ⊂ H^{2} of g–self–dual harmonic forms along the subspace H_{−}^{2} of g–anti–self–dual harmonic forms. The argument above really proves

(c_{1}(P̃(X))^{+})^{2} ≤ \frac{1}{32π^{2}} ∫_{X} s_{g}^{2}dvol_{g} .

If P̃(Y) is a Spin^{c}–structure on Y with non–zero Seiberg–Witten invariant, then, by Proposition 1 and the subsequent remark, there are Spin^{c}–structures P̃(X) on X = Y #k \overline{ℂP}^{2} with non–zero Seiberg– Witten invariants and with

c_{1}(P̃(X)) = c_{1}(P̃(Y)) + ∑_{i=1}^{k}(−1)^{ϵ_{i}}E_{i}

for any choice of the signs (−1)^{ϵ_{i}}. Choose the signs so that

(−1)^{ϵ_{i}}E_{i}^{+} · c_{1}(P̃(Y))^{+} ≥ 0 .

Then

\frac{1}{32π^{2}} ∫_{X} s_{g}^{2}dvol_{g} ≥ (c_{1}(P̃(X))^{+})^{2}

= (c_{1}(P̃(Y))^{+})^{2} + 2 ∑_{i=1}^{k}(−1)^{ϵ_{i}}E_{i}^{+} · c_{1}(P̃(Y))^{+} + (∑_{i=1}^{k}(−1)^{ϵ_{i}}E_{i}^{+})^{2}

≥ (c_{1}(P̃(Y))^{+})^{2} ≥ c_{1}(P̃(Y))^{2} ≥ 2e(Y) + 3σ(Y)

= 2(e(X) − k) + 3(σ(X) + k) = 2e(X) + 3σ(X) + k ,

where we have used the inequality c_{1}(P̃(Y))^{2} ≥ 2e(Y)+3σ(Y) which is equivalent to the assertion that the moduli space associated with P̃(Y) has non–negative dimension.

Thus, we have proved \frac{1}{32π^{2}} ∫_{X} s_{g}^{2}dvol_{g} ≥ 2e(X) + 3σ(X) + k for every metric g on X.

Suppose now that g is Einstein. Then the Chern–Weil integrals for the Euler characteristic and the signature of X give

2e(X) + 3σ(X) = \frac{1}{4π^{2}} ∫_{X}(\frac{1}{24}s_{g}^{2} + 2|W_{+}|^{2})dvol_{g}

≥ \frac{1}{96π^{2}} ∫_{X} s_{g}^{2}dvol_{g}

≥ \frac{1}{3}(2e(X) + 3σ(X) + k) ,

where W_{+} denotes the self–dual part of the Weyl tensor of g. Therefore k ≤ 2(2e(X) + 3σ(X)) = 2(2e(Y) + 3σ(Y) − k), which implies k ≤ \frac{2}{3}(2e(Y) + 3σ(Y)).

Theorem 2 was proved by LeBrun [19], who also discussed the borderline case k = \frac{2}{3}(2e(Y) + 3σ(Y)), in the case where Y is complex or symplectic. In that case the blown up manifold X is also complex, respectively symplectic, so that Proposition 1 is not needed. Otherwise the argument is the same. For our application below, the result of [19] would be sufficient.

The following is the main result of this section, giving the examples mentioned in the introduction.

Theorem 3. There are infinitely many pairs (X_{i}, Z_{i}) of simply connected closed oriented smooth 4–manifolds such that:

1. X_{i} is homeomorphic to Z_{i},

2. if i ≠ j, then X_{i} and X_{j} are not homotopy equivalent,

3. Z_{i} admits an Einstein metric but X_{i} does not,

4. e(X_{i}) > \frac{3}{2}|σ(X_{i})|.

Note that 3. implies in particular that X_{i} and Z_{i} are not diffeomorphic.

Proof. We claim that there are simply connected minimal complex surfaces Y_{i}, Z_{i} of general type such that if we take X_{i} = Y_{i}#k\overline{ℂP^{2}}, for a suitable k with k > \frac{2}{3}(2e(Y_{i}) + 3σ(Y_{i})), then the pairs (X_{i}, Z_{i}) have all the desired properties. The last property, the strict Hitchin–Thorpe inequality, follows from the Noether and Miyaoka–Yau inequalities for Z_{i}, which, by the first property, has the same Euler characteristic and signature as X_{i}.

If we take Z_{i} to have ample canonical bundle, then the results of Aubin and Yau on the Calabi conjecture show that Z_{i} admits a Kähler– Einstein metric, compare [3]. On the the other hand, X_{i} does not admit any Einstein metric by Theorem 2.

The crucial issue then is to arrange that Z_{i}, with ample canonical bundle, is homeomorphic to the k–fold blowup of Y_{i}, with k > \frac{2}{3}(2e(Y_{i}) + 3σ(Y_{i})). As X_{i} will be automatically non–spin, X_{i} and Z_{i} will be homeomorphic by Freedman’s classification [10] as soon as Z_{i} is non–spin and has the same Euler characteristic and the same signature as X_{i}. One can find suitable surfaces using the known results on the geography of surfaces of general type, see [15] for a summary of the results.

To exhibit concrete examples, instead of working with the topological Euler characteristic and the signature, we shall use the first Chern number c_{1}^{2} = 2e+3σ and the Euler characteristic of the structure sheaf χ = \frac{1}{4}(e + σ).

Under blowing up, c_{1}^{2} drops by one and χ is constant. Thus, the Miyaoka–Yau inequality for Y_{i} implies c_{1}^{2}(X_{i}) < 3χ(X_{i}). In fact, c_{1}^{2}(X_{i}) will be smaller still, because simply connected surfaces Y_{i} are not known to exist if we get too close to the Miyaoka–Yau line c_{1}^{2} = 9χ.

The minimal surface Z_{i} satisfies the same inequalities on its characteristic numbers as X_{i}. If the canonical bundle of Z_{i} is very ample, then Castelnuovo’s theorem, see [2] page 228, gives c_{1}^{2}(Z_{i}) ≥ 3χ(Z_{i}) − 10, which will contradict the above upper bound for c_{1}^{2}(X_{i}). Thus, Z_{i} must be chosen to have ample but not very ample canonical bundle, and will be in the sector where

2χ(Z_{i}) − 6 ≤ c_{1}^{2}(Z_{i}) < 3χ(Z_{i}) ,

the first being the Noether inequality. The results of Xiao Gang and Z. Chen, cf. [15], show that all non–spin simply connected surfaces Z_{i} with ample canonical bundle which are in this sector, and not too close to the line c_{1}^{2} = 3χ, will have companions X_{i} as required, obtained by blowing up minimal surfaces Y_{i}. Note that by Beauville’s theorem on the canonical map, cf. [2] page 228, all the Z_{i} will be double covers of ruled surfaces.

We can avoid using the results of Xiao and Chen by taking for Z_{i} the following family of Horikawa surfaces, cf. [2]. Let Σ_{i} be the Hirzebruch surface whose section at infinity S has selfintersection −i, and let Z_{i} be a double cover of Σ_{i} branched in a smooth curve homologous to B = 6S + 2(2i + 3)F, where F is the class of the fiber. The double cover is simply connected as B is ample, and K_{Z_{i}} = π^{∗}(K_{Σ_{i}} + \frac{1}{2}B) = π^{∗}(S + (i + 1)F) is not 2–divisible and so Z_{i} is not spin. Moreover, K_{Z_{i}} is the pullback of an ample line bundle and therefore ample, so that Z_{i} admits a Kähler–Einstein metric. The characteristic numbers of Z_{i} are c_{1}^{2}(Z_{i}) = 2i + 4 and χ(Z_{i}) = i + 5.

Now, by the classical geography results of Persson [20], for all i large enough there are simply connected surfaces Y_{i} of general type with c_{1}^{2}(Y_{i}) = 6i + 13 and χ(Y_{i}) = i + 5, so that the (4i + 9)–fold blowup X_{i} of Y_{i} is homeomorphic to Z_{i}.

The pairs (X_{i}, Z_{i}) have all the desired properties.

Remark 1. The examples of manifolds without Einstein metrics given by LeBrun [19], namely blowups of hypersurfaces in ℂP^{3}, cannot be used to prove Theorem 3 because they violate the Noether inequality. They are therefore not homeomorphic to minimal surfaces for which the resolution of the Calabi conjecture gives existence of an Einstein metric.

In view of Theorem 3, one can ask how many smooth structures with Einstein metrics and how many without a given topological manifold has. On the one hand, using for example the work of Fintushel–Stern, one can show that one has infinitely many choices for the smooth structures of the manifolds Y_{i} in the proof of Theorem 3, which remain distinct under blowing up points. Thus, one has infinitely many smooth manifolds one can use for each X_{i}, not admitting any Einstein metrics. On the other hand, I do not know a single example of two surfaces with ample canonical bundles which are homeomorphic but not diffeomorphic, giving a choice of two different smooth structures for the Z_{i} in the proof of Theorem 3. However, we will see in Theorem 5 below that there are 4–manifolds admitting Einstein metrics for more than one smooth structure.

2. Uniqueness for a given smooth structure

We have seen that existence of Einstein metrics on closed 4–manifolds depends in an essential way on the smooth structure. I believe that the issue of uniqueness is also tied to the smooth structure. More specifically:

Conjecture 4. A closed smooth 4–manifold admits Einstein metrics for at most one sign of the scalar curvature.

All the known examples, and the results of Seiberg–Witten theory, support this conjecture. The conjecture is interesting because it is sharp – it would be false if one did not fix the smooth structure, but only the underlying topological manifold:

Theorem 5. There are simply connected homeomorphic but non- -diffeomorphic smooth 4–manifolds X and Y, such that X admits an Einstein metric of positive scalar curvature, and Y admits an Einstein metric of negative scalar curvature.

Proof. We can take for X the 8–fold blowup of ℂP^{2}. By the work of Tian–Yau [21] this admits a Kähler–Einstein metric of positive scalar curvature.

For Y we take the simply connected numerical Godeaux surface constructed by Craighero–Gattazzo [6] and studied recently by Dolgachev– Werner [7]^{2}. This has ample canonical bundle, so that by the work of Aubin and Yau it admits a Kähler–Einstein metric of negative scalar curvature.

By Freedman’s classification [10], X and Y are homeomorphic. That they are not diffeomorphic is clear from [14]. The argument carried out there for the Barlow surface, cf. [1], works even more easily for the Craighero–Gattazzo surface as there is no complication arising from (−2)–curves. Alternatively, the fact that X and Y have Kähler– Einstein metrics of opposite signs implies via Seiberg–Witten theory that they are non–diffeomorphic.

In higher dimensions, homeomorphic non–diffeomorphic manifolds with Einstein metrics are known [18, 22], though in those examples all the metrics have positive scalar curvature. Although the examples of [18, 22] are consistent with a higher–dimensional analogue of the above conjecture, such a generalisation is false:

Corollary 6. For every i ≥ 2 there is a simply connected closed 4i– manifold which admits Einstein metrics of both positive and negative scalar curvature.

Proof. Let X and Y be as in Theorem 5. Then the i–fold products X^{i} = X × … × X and Y^{i} = Y × … × Y have Kähler–Einstein metrics of positive, respectively negative, scalar curvature. However, as X and Y are simply connected and homeomorphic, they are h–cobordant. Therefore, X^{i} and Y^{i} are also h–cobordant for all i, and for i ≥ 2 are diffeomorphic by the h–cobordism theorem.

Theorem 5 and the Corollary have also been proved independently by Catanese and LeBrun [5]. Instead of the Craighero–Gattazzo surface they use the Barlow surface, showing that it has deformations with ample canonical bundle. Conjecturally, the Barlow and Craighero– Gattazzo surfaces are deformation equivalent, and therefore diffeomorphic.

2I am grateful to Igor Dolgachev for telling me about this, and for providing an advance copy of [7].

References

1. R. Barlow, A simply connected surface of general type with p_{g} = 0, Invent. Math. 79 (1985), 293–301.

2. W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer Verlag 1984.

3. A. L. Besse, Einstein Manifolds, Springer Verlag 1987.

4. G. Besson, G. Courtois et S. Gallot, Entropies et rigidités des espaces localement symmetriques de courbure strictement négative, Geom. and Funct. Analysis 5 (1995), 731–799.

5. F. Catanese and C. LeBrun, On the scalar curvature of Einstein manifolds, preprint May 1997.

6. P. Craighero and R. Gattazzo, Quintic surfaces of ℂP^{3} having a nonsingular model with q = p_{g} = 0, P_{2} ≠ 0, Rend. Sem. Mat. Univ. Padova 91 (1994), 187–198.

7. I. Dolgachev and C. Werner, A simply connected numerical Godeaux surface with ample canonical class, preprint April 1997.

8. S. K. Donaldson, The Seiberg–Witten equations and 4–manifold topology, Bull. Amer. Math. Soc. 33 (1996), 45–70.

9. R. Fintushel and R. J. Stern, Immersed spheres in 4–manifolds and the immersed Thom conjecture, Proc. of Gökova Geometry–Topology Conference 1994, Turkish J. of Math. 19 (2) (1995), 27–39.

10. M. H. Freedman, The topology of four–manifolds, J. Differential Geometry 17 (1982), 357–454.

11. R. Friedman and J. W. Morgan, Smooth Four–Manifolds and Complex Surfaces, Springer Verlag 1994.

12. M. Gromov, Volume and bounded cohomology, Publ. Math. I.H.E.S. 56 (1982), 5–99.

13. N. J. Hitchin, Compact four–dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435–441.

14. D. Kotschick, On manifolds homeomorphic to ℂP^{2}#8\overline{ℂP^{2}}, Invent. math. 95 (1989), 591–600.

15. D. Kotschick, Orientation–reversing homeomorphisms in surface geography, Math. Annalen 292 (1992), 375–381.

16. D. Kotschick, P. B. Kronheimer and T. S. Mrowka, monograph in preparation.

17. D. Kotschick, J. W. Morgan and C. H. Taubes, Four–manifolds without symplectic structures but with non–trivial Seiberg–Witten invariants, Math. Research Letters 2 (1995), 119–124.

18. M. Kreck and S. Stolz, A diffeomorphism classification of 7–dimensional homogeneous Einstein manifolds with SU(3) × SU(2) × U(1) symmetry, Ann. of Math. 127 (1988), 373–388.

19. C. LeBrun, Four–manifolds without Einstein metrics, Math. Research Letters 3 (1996), 133–147.

20. U. Persson, Chern invariants of surfaces of general type, Comp. Math. 43 (1981), 3–58.

21. G. Tian and S.-T. Yau, Kähler–Einstein metrics on complex surfaces with c_{1} > 0, Comm. Math. Phys. 112 (1987), 175–203.

22. McK. Y. Wang and W. Ziller, Einstein metrics on principal torus bundles, J. Differential Geometry 31 (1990), 215–248.

23. E. Witten, Monopoles and four–manifolds, Math. Research Letters 1 (1994), 769–796.

Max–Planck–Institut für Mathematik, Gottfried–Claren–Str. 26, 53225 Bonn, Germany