Small bound for birational automorphism groups of algebraic varieties (with an Appendix by Yujiro Kawamata)

We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n =  3) of general type in terms of the volume V =  V(X) under an “albanese smoothness and simplicity” condition. To be precise, $$|{\rm Bir}(X)| \leq d_3 V^{10}$$ . An optimum linear bound $$|{\rm Bir}(X)| \leq \frac{1}{3} \times 42^3 V$$ is obtained for those threefolds with non-maximal albanese dimension. For all n ≥ 3, a bound $$|{\rm Bir}(X)| \leq d_n V^{10}$$ is obtained when albX is generically finite, alb(X) is smooth and Alb(X) is simple.