# 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?

• maybe you can check numdam.org/item?id=CM_1987__63_1_123_0 Sep 8, 2018 at 5:16
• @ChenJiang Thank you, I think Theorem 3.10 in the reference you gave would make it! Sep 8, 2018 at 21:02

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. Sep 8, 2018 at 11:57
• And I think you missed the part "and at the references cited therein". Sep 8, 2018 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, , , , , , , and  for surfaces;, , , , and  in higher dimensions; and  for surfaces in characteristic $p$". Matsumura's article is . Sep 8, 2018 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 ). You can read page 1079 line -5 of HMX paper. Sep 8, 2018 at 12:37
• Ok, I see. I will provide an alternative reference. Sep 8, 2018 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.