# Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero.

Then Hurwitz proved that the number of automorphisms of $X$ is at most $84(g-1)$.

I would like to know why Aut$(X)$ is finite (without appealing to Hurwitz' result) by an "elementary" argument.

That is, why should the automorphism group of $X$ be finite?

Once I have such an elementary argument, I believe it should be applicable to certain higher-dimensional varieties such as varieties with ample canonical sheaf. Why should they have only finitely many automorphisms?

-
I heard of two proofs. The first one imply calculating volumes of fundamental polygon of the curve in the universal cover. The second is using Weiestrass points (which are in finite number) because the group of automorphisms acts on the Weiestrass points. – Damien L Feb 7 '13 at 0:23
I don't know an elementary argument.To deal with higher dimension case, one can consider $\mathrm{Aut}(X)$ as an algebraic group. Its tangent space at the origin is a vector space of dimension $\dim H^0(X, T_X)$ where $T_X$ is the tangent bundle of $X$. So the whole reduces to show the vanishing of $H^0(X, T_X)$. Note that the finiteness of $\mathrm{Aut}(X)$ is not true in positive characteristic when $\dim X>1$. – user18119 Feb 7 '13 at 21:58
@user18119. I'm afraid but the condition $h^0(X, \Theta _X) = 0$ does not imply $Aut(X)$ to be finite: There exist K3-surfaces $X$ with $Aut(X) \cong \mathbb Z_2 * \mathbb Z_2$ an infinite non-Abelian group. On the other hand, $h^0(X, \Theta _X) = 0$ for each K3-surface $X$. Note: From $h^0(X, \Theta _X) = 0$ one can only conclude that the complex Lie group $Aut(X)$ is zero-dimensional. – Jo Wehler Oct 19 '14 at 11:37

All compact complex surfaces $X$ of general type have a finite group $Aut(X)$ of automorphism. If $X$ is minimal, even numerical bounds are known for the number $\#Aut(X)$, e.g. a bound linear in $c_1^2(X)$. Cf. "Xiao, Gang: Bound of automorphisms of surfaces of general type, I. Ann. of Math. (139) 1994, 51-77"

-

@Tom, because you are interested in a generalization to higher dimensional varieties.

To prove the finiteness of $G := Aut(X)$ for a surface $X$ of general type one can proceed as follows: By definition $X$ has Kodaira dimension $kod(X) = 2$. The plurigenera - and the Kodaira dimension in particular - are birational invariants. A fundamental result about surfaces of general type states: The 5-canonical map

$$\phi_{\kappa_X^{\otimes 5}}: X \longrightarrow \mathbb P^N$$

maps $X$ birationally onto a normal surface $Z \subset \mathbb P^N, N = h^0(X, \kappa_X^{\otimes 5}) - 1$.

This fact allows to represent the complex Lie group $G$ as an affine algebraic group: The natural action of $G$ on $X$ induces an action on the vector space $H^0(X, \kappa_X^{\otimes 5})$. Because $G$ stabilizes the closed subvariety $Z \in \mathbb P^N$, $G$ is a closed subgroup of the projective linear group. The latter is an affine algebraic group, hence $G$ is an affine algebraic group, too.

If $G$ were not finite, a 1-dimensional subgroup $A \subset G, A = \mathbb G_m \ or \ A = \mathbb G_a$, would exist. The group $A$ is Abelian. The orbit space $Z/A$ exists as a variety with a rational map $f: Z \longrightarrow Z/A$. By the 1-dimensional case of the "cross-section theorem" of Rosenlicht $f$ has a rational section, which implies that $Z$ is birational to a ruled surface. Any ruled surface - and a posteriori $Z$ - has Kodaira dimension = $- \infty$. This fact contradicts $kod(X) = 2$. Hence $G$ is finite, q.e.d.

Cf. "Rosenlicht, Maxwell: Some Basic Theorems on Algebraic Groups. American Jounal of Mathematics, Vol. 78, N0. 2 (1956), Theorem 10."

-
Nice answer. This also holds in higher dimension, right ? – Cantlog Oct 25 '14 at 6:17
@Cantlog: Yes, the statement also holds in higher dimensions. For an estimation with bounds see "Christopher D. Hacon, James McKernan, Chenyang Xu: On the birational automorphisms of varieties of general type. Annals of Mathematics, 177 (2013), 1077-1111. – Jo Wehler Oct 26 '14 at 17:44