Euler-Characteristic and signature of Kähler hyperbolic manifolds

Thomas Schick^{∗} Fachbereich Mathematik — Uni Münster Einsteinstr. 62, D-48149 Münster

March 17, 1998

Abstract We combine Atiyah’s L^{2}-index theorem and Gromov’s results on the L^{2}-cohomology of

Kähler hyperbolic manifolds with rather trivial observations to conclude:

If M^{2k} is a Kähler hyperbolic manifold of real dimension 2k, then (−1)^{k}χ(M) ≥ k + 1, and for the signature we have |sgn(M)| ≤ χ(M) − k.

First, we give a reminder on Kähler hyperbolic manifolds as introduced by Gromov [3].

1. Definition. A Kähler hyperbolic manifold is a compact Kähler manifold, so that the lift ω̃ of the Kähler form ω to the universal covering M̃ is d̃(bounded), that means there is a bounded 1-form η on M̃ with ω = dη.

Many Kähler manifolds are in fact Kähler hyperbolic, f.i. [3]:

• Compact Kähler manifolds which are homotopy equivalent to compact Riemannian manifolds with negative sectional curvature.

• Compact Kähler manifolds with word hyperbolic fundamental group (in the sense of Gromov [4]) and vanishing π_{2}

• Products and complex submanifolds of Kähler hyperbolic manifolds

The main result for Kähler hyperbolic manifolds is Gromov’s computation of their L^{2}-cohomology [3, 2.5]:

2. Theorem. Let M be a Kähler hyperbolic manifold of real dimension 2m with universal covering M̃. Then (1) H_{(2)}^{p,q}(M̃) = 0 for p + q ≠ m, in particular b_{r}^{(2)} = 0 if r ≠ m.

∗e-mail: thomas.schick@math.uni-muenster.de

(2) H_{(2)}^{p,q}(M̃) ≠ 0 for p + q = m, in particular b_{m}^{(2)} ≠ 0.

3. Remark. The first statement establishes the Singer conjecture [2, Conjecture 2] for Kähler hyperbolic manifolds. Gromov’s method was extended by Jost/Zuo[5, Theorem 3] to Kähler nonelliptic manifolds, f.i. Kähler manifolds with a metric with nonpositive sectional curvature [5, p. 4] to get assertion (1) of Theorem 2.

4. Remark. The second statement of Theorem 2 together with Atiyah’s L^{2}-index theorem [1] implies the Hopf conjecture [7, 2.7] (−1)^{m}χ(M) > 0 for Kähler hyperbolic manifolds.

Our aim is to show that one can strengthen this in the following way:

5. Proposition. Let M be a Kähler hyperbolic manifold of real dimension 2m. Then

(−1)^{m}χ(M) ≥ m + 1.  (6)

Moreover, the following relation between Euler characteristic and signature holds if m is even:

|sgn(M)| ≤ χ(M) − m.  (7)

8. Remark. Lück [6, 0.7] uses L^{2}-cohomology to conclude a similar inequality between signature and Euler characteresitic for certain 4-manifolds.

To prove the proposition, we want to estimate the von Neumann dimensions of H_{(2)}^{p,q}. Set π := π_{1}(M). H_{(}^{p,q}2) are Hilbert modules over the group von Neumann algebra N (π). The L^{2}-index theorem implies

χ(M) = ∑_{p} dim_{π}(−1)^{p}H_{(2)}^{p}(M̃).

From Hodge theory we know:

H^{k} = ⨁_{p+q=k} H_{(2)}^{p,q}.

At first sight, Gromov’s Theorem 2 does not imply more than the nonvanishing of the Euler characteristic because the von Neumann dimension can be an arbitrary nonnegative real number, a priori. We will show that dim_{π} H_{(2)}^{p,q} is the index of an operator and therefore integral for every p + q = m. To do this, we employ Atiyah’s L^{2}-index theorem again: Study the differential operator ¯∂_{p} + ¯∂_{p}^{∗} : Ω^{p,even} → Ω^{p,odd}. Its square is the elliptic differential operator Δ/2, therefore ¯∂_{p} + ¯∂_{p}^{∗} is elliptic. This implies that we can apply the L^{2}-index theorem to this operator:

ℤ ∋ ind(¯∂_{p} + ¯∂_{p}^{∗}) = ind_{π}(˜¯∂ + ˜ ¯∂^{∗}) = ∑_{q even} dim_{π} H_{(2)}^{p,q} − ∑_{q odd} dim_{π} H_{(2)}^{p,q} =^{Theorem 2} (−1)^{m−p} dim_{π} H_{(2)}^{p,m−p}

⟹ dim_{π} H_{(2)}^{p,m−p} ∈ ℕ ⟹^{H_{(2)}^{p,m−p}≠0} dim_{π} H_{(2)}^{p,m−p} ≥ 1.  (9)

Therefore

(−1)^{m}χ(M) = dim_{π} H_{(2)}^{m}(M̃) = ∑_{p=0}^{m} dim_{π} H_{(2)}^{p,m−p}(M̃) ≥ m + 1.

For the signature remember that if m is even, then we split ∑_{p,q} Ω^{p,q} into the ±1 eigenspaces of the Hodge-∗ operator. The signature is the index of d + d^{∗} with respect to this splitting. Using the L^{2}-index theorem again, and Gromov’s result, we have therefore

sgn(M) = dim_{π}(1 + ∗)H_{(2)}^{m}(M̃) − dim_{π}(1 − ∗)H_{(2)}^{m}(M̃)

= ∑_{p=0}^{m} dim_{π}(1 + ∗)H_{(2)}^{p,m−p}(M̃) − ∑_{p=0}^{m} dim_{π}(1 − ∗)H_{(2)}^{p,m−p}(M̃).

Now remember that ∗ : H^{p,q} → H^{m−p,m−q} is an isomorphism. It follows that projection (1 ± ∗)H^{p,m−p} → H^{p,m−p} is an (π-invariant) isomorphism if p ≠ m − p, i.e. if p ≠ m/2. Therefore most of the terms in the sum cancel, and it remains:

|sgn(M)| = |dim_{π}(1 + ∗)H_{(2)}^{m/2,m/2}(M̃) − dim_{π}(1 − ∗)H_{(2)}^{m/2,m/2}(M̃)|

≤ dim_{π} H_{(2)}^{m/2,m/2}(M̃) = χ(M) − ∑_{p≠m/2} dim_{π} H_{(2)}^{p,m−p}(M̃)

≤^{(9)} χ(M) − m.

This finishes the prove of (7).

References

[1] Atiyah, M.: “Elliptic operators, discrete groups and von Neumann algebras”, Astérisque 32, 43–72 (1976)

[2] Dodziuk, J.: “L^{2}-harmonic forms on rotationally symmetric Riemannian manifolds”, Proc. of the AMS 77, 395–400 (1979)

[3] Gromov, M.: “Kähler hyperbolicity and L_{2}-Hodge theory”, Journal of Differential Geometry 33, 263–292 (1991)

[4] Gromov, M.: “Asymptotic invariants of infinite groups”, in: Niblo, G.A. and Roller, M.A. (eds.), Geometric groups theory, volume 2, Proc. of the Symp. in Sussex 1991 , vol. 182 of Lecture Notes series, Cambridge University Press (1993)

[5] Jost, J. and Zuo, K.: “Vanishing theorems for L^{2}-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry”, preprint (1997)

[6] Lück, W.: “Hilbert modules and modules over finite von Neumann algebras and applications to L^{2}-invariants”, Mathematische Annalen 309, 247–285 (1997)

[7] Lück, W.: “L^{2}-invariants of regular coverings of compact manifolds and CW -complexes”, to appear in “Handbook of Geometry”, Elsevier (1997)