# Reference request: birational automorphism group is finite

I am interested in having a look at the proof of the following fact: If $X$ is a smooth variety of general type, then $\mathrm{Aut(X)}$ is finite.

I know that this is proved in "On algebraic groups of birational transformations" by Matsumura. Unfortunately, this paper is not available at the library of my institution, nor I could find it online. Could anyone point at a way to find such paper, or an alternative reference?

EDIT. An alternative reference is

Husemoller, Dale H.: Finite automorphism groups of algebraic varieties, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 611-619 (1980). ZBL0466.14018.

A lot of work has been done in order to bound $|\mathrm{Aut}(X)|$ in terms of the dimension and the volume of $X$. You can look at

Hacon, Christopher D.; McKernan, James; Xu, Chenyang: On the birational automorphisms of varieties of general type, Ann. Math. (2) 177, No. 3, 1077-1111 (2013). ZBL1281.14036

and at the references cited therein. Quoting from the abstract:

We show that the number of birational automorphisms of a variety of general type $X$ is bounded by $c⋅\mathrm{vol}(X,\, K_X)$, where $c$ is a constant that only depends on the dimension of $X$.

• I think you missed his point. In the paper of HMX they actually used the fact that the automorphism group is finte, which goes back to Matsumura. Hanamura’s paper also proved this. – Chen Jiang Sep 8 '18 at 11:57
• And I think you missed the part "and at the references cited therein". – Francesco Polizzi Sep 8 '18 at 11:59
• At page 1078 (the second one of the paper) you can read "This problem has been extensively studied in higher dimensions; see, for example, [1], [3], [8], [10], [16], [30], and [31] for surfaces;[9], [26], [29], [32], and [33] in higher dimensions; and [5] for surfaces in characteristic $p$". Matsumura's article is [22]. – Francesco Polizzi Sep 8 '18 at 12:04
• By "this problem", they mean "how large this group is", for example, bounded by a polynomial function of the volume. The finiteness is proved by Matsumura (which is known earlier that the above references except [3]). You can read page 1079 line -5 of HMX paper. – Chen Jiang Sep 8 '18 at 12:37
• Ok, I see. I will provide an alternative reference. – Francesco Polizzi Sep 8 '18 at 13:09

See $\S$11.7 or more specifically Theorem 11.12 in Algebraic Geometry An Introduction to Birational Geometry of Algebraic Varieties by Shigeru Iitaka.