Hirzebruch’s Proportionality

September 30, 1999

(References) F. Hirzebruch, Characteristic numbers of homogeneous domains, (1957). F. Hirzebruch, Automorphic formen und der Satz von Riemann-Roch, (1958).

1. Let X be a complex manufold of dimension n. The total Chern class of X is given by

c(X) = 1 + c_{1} + c_{2} + ⋯ + c_{n} ≅ det(I + \frac{\sqrt{−1}}{2π}Θ),

where Θ is the curvature matrix of X, i.e., Θ is a local expression of the curvature R of X.

2. Let X be compact. For any partition λ = (i_{1}, i_{2}, ⋯, i_{r}) of n, a Chern number c_{λ}(X) ∈ H^{2n}(X, Z) is defined by the relation

c_{λ} · h_{X} = c_{i_{1}}c_{i_{2}} ⋯ c_{i_{r}},

where h_{X} is the generator of H^{2n}(X, Z).

3. Two compact complex manifolds X and X̃ of dimension n are called proportional if for any two partitions λ and μ of n, the following proportionality holds:

\frac{c_{λ}(X)}{c_{μ}(X)} = \frac{c_{λ}(X̃)}{c_{μ}(X̃)}.

For example, any two compact manifolds of dimension n are proportional if there is a local diffeomorphism.

X: noncompact irreducible Hermitian symmetric space, X̃: the compact dual Hermitian symmetric space, Γ: discrete subgroup of Aut(X) acting on X without fixed points. ( ⇒ Γ\X is a compact manifold of dimension n.)

\underline{Theorem} (Hirzebruch) X̃ and Γ\X are proportional.

Compact irreducible Hermitian symmetric spaces

SU(p + q)/S(U(p) × U(q)), Grassmannians SO(2n)/U(n), maximal orthogonal Grassmannians SP(n)/U(n), Lagrangian Grassmannians SO(n + 2)/(SO(2) × SO(n)), quadric hypersurfaces E_{6}/Spin(10) × T^{1} E_{7}/E_{6} × T^{1}

Hirzebruch’s proportionality gives information on the Chern numbers of the compact quotients of bounded symmetric domains:

\frac{c_{λ}(Γ\X)}{c_{n}(Γ\X)} = \frac{c_{λ}(X̃)}{c_{n}(X̃)},

hence

c_{λ}(Γ\X) = \frac{χ(Γ\X)}{χ(X̃)}c_{λ}(X̃).

Let P^{2} = P^{2}(C), which is dual to the ball B^{2}. Then,

c(P^{2}) = (1 + H)^{3} = 1 + 3H + 3(H^{2}),

where H is the hyperplane of P^{2} which generates H^{2}(P^{2}, Z). Hence c_{1}^{2} = 3c_{2} and this relation holds for any surface given by M = Γ\B^{2} by Hirzebruch’s proportionality.

\underline{Theorem} (Yau, 1977) For a Kähler surface M whose canonical line bundle is ample,

(i) 3c_{2} ≥ c_{1}^{2}. (ii) The equality holds if and only if M is covered by B^{2}. Higher dimensional analogue.

(Sketch of proof)

X: bounded symmetric domain ⊂ C X̃: compact dual manifold Γ\X: compact quotient

1. \underline{X and Γ\X}.

We must define the Chern numbers of X (noncompact). The Bergmann metric of X is given by

ds^{2} = ∑ \frac{∂^{2}logK}{∂z_{α}∂z̄_{β}} dz_{α} ⊗ d̄z_{β} = ∑ g_{αβ̄}dz_{α} ⊗ d̄z_{β},

where K is the Bergmann’s kernel function. We have the associates closed (1,1)-form

ω = \frac{\sqrt{−1}}{2π} ∑ g_{αβ̄}dz_{α} ∧ d̄z_{β}

and ω^{n}/n! is the real volume element.

For any partition λ = (i_{1}, i_{2}, ⋯, i_{r}) of n, the 2n-form c_{i_{1}}c_{i_{2}} ⋯ c_{i_{r}} is a multiple of the volume element ω^{n}/n!. The Chern number c_{λ} of X is defined by the relation

c_{λ} · \frac{ω^{n}}{n!} = (−1)^{n}c_{i_{1}}c_{i_{2}} ⋯ c_{i_{r}}.

Since the covering map p : X → Γ\X is a local isometry, we get

p^{∗}(c_{i}(Γ\X)) = c_{i}(X), ∀ i.

Hence it is clear that X and Γ\X are proportional.

2. \underline{X̃ and X}.

X̃ = G/H, 𝔤 and 𝔥: Lie algebras of G and H. Cartan decomposition 𝔤 = 𝔥 + 𝔪, where 𝔪 = T_{e}X. C(G) : the complex extension of G

with Lie algebra 𝔠 = 𝔥 + i𝔥 + 𝔪 + i𝔪. G̃ ≤ C(G) : the Lie subgroup

corresponding to 𝔤̃ = 𝔥 + i𝔪 ⊂ 𝔠.

The space X̃ = G̃/H is just the compact dual of X. We get the isomorphism of the tangent spaces:

T_{e}X = 𝔪 ≅ i𝔪 = TẽX̃,
v ↦ iv.

For symmetric spaces X and X̃, the curvature tensor is given by

R(x, y)z = −[[x, y], z], x, y, z ∈ 𝔪 (or i𝔪),

hence

R̃(ix, iy)(iz) = i[[x, y], z] = −iR(x, y)z.

Recall that the total Chern class is given by

c(X) = det(I + \frac{\sqrt{−1}}{2π}R).

These two formulas shows that X and X̃ are proportional.

From 1. and 2., we conclude that X̃ and Γ\X are proportional. □