A Hopf manifold is a compact complex manifold that can be obtained as a quotient of the punctured complex Euclidean space $\mathbb{C}^n \setminus {0}$ by a properly discontinuous group of dilations. The most common construction uses a single linear contraction:
$$ \Gamma = \langle A\rangle,\qquad A(z)=\alpha z,\ \ 0<|\alpha|<1, $$
where $A$ acts on $\mathbb{C}^n$ by complex scalar multiplication. The Hopf manifold is then
$$ H = (\mathbb{C}^n \setminus {0}) / \Gamma . $$
For $n=2$ the resulting manifolds are called Hopf surfaces; for $n\ge 2$ they are higher‑dimensional Hopf manifolds.
Topological and differentiable structure
- As a smooth manifold, a Hopf manifold is diffeomorphic to the product $S^{1}\times S^{2n-1}$. The map $$ \mathbb{C}^n \setminus {0}\longrightarrow S^{1}\times S^{2n-1},\qquad z\mapsto \bigl(\arg(z), z/|z|\bigr) $$ descends to a diffeomorphism after passing to the quotient.
- Consequently, the fundamental group of a Hopf manifold is infinite cyclic, $\pi_1(H)\cong\mathbb{Z}$.
Complex-analytic properties
- Hopf manifolds are complex manifolds of complex dimension $n$ that are non‑Kähler; their first Betti number $b_1 = 1$ violates the Kähler condition $b_1$ even.
- Their Hodge numbers satisfy: $$ h^{0,0}=h^{n,n}=1,\qquad h^{1,0}=h^{0,1}=0,\qquad h^{p,q}=0\ \text{for}\ (p,q) eq(0,0),(n,n). $$
- They have Kodaira dimension $-\infty$.
Generalizations
- Primary Hopf manifolds are defined as above using a single contraction.
- Secondary Hopf manifolds arise when the acting group $\Gamma$ is generated by several commuting linear maps that are simultaneously diagonalizable with eigenvalues of modulus less than one.
- More generally, any compact complex manifold whose universal cover is $\mathbb{C}^n\setminus{0}$ and whose deck transformation group is infinite cyclic is a Hopf manifold.
Historical notes
The class was introduced by Heinz Hopf in the 1940s in the context of classifying complex surfaces. Hopf surfaces (the case $n=2$) were among the first examples of compact complex manifolds that are not Kähler.
References
- H. Hopf, “Über die Abbildungen der komplexen Ebene auf sich selbst,” Mathematische Annalen 96 (1927): 445–456.
- A. Fujiki, “On automorphism groups of compact Kähler manifolds.” Inventiones Mathematicae 44 (1978): 225–258.
- K. Kodaira, “On the structure of compact complex analytic surfaces. I.” American Journal of Mathematics 86 (1964): 751–798. (Provides classification of Hopf surfaces.)
