TITLE: When does $Aut(X)=Bir(X)$ hold?
QUESTION [12 upvotes]: Let $X$ be a projective complex manifold. Under what condition do we have the equality $Aut(X)=Bir(X)$? Here $Aut(X)$ denotes the group of holomorphic automorphisms of $X$ and $Bir(X)$ the group of birational morphisms of $X$.
I am interested in the case when $\dim_{\mathbb{C}}X=2,3$. Maybe there are not universal criteria, so I would appreciate your providing me with any examples for which the equality holds.

REPLY [9 votes]: A simplest example is a curve $C$, then $Bir(C)\cong Aut(C)$
A careful note that $Bir(X)$ is not a group scheme in general.Moreover if $X$ and $Y$ are birationl then $Bir(X)$ is not isometric with $Bir(Y)$
So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $Bir(X)$ good behave since we have luck of well defined multiplication rule. In fact is $X$ be a minimal model with terminal singularities it is a theorem that $Bir(X)$ is a group scheme as an Abelian variety. In general $$Aut^o(X)\cong Bir^o(X)$$
where $Aut^o$ is the identity connected component. See this paper 
Let $X$ be a projective variety then we have the following isomorphism from MMP
$$Aut(X_{min})\cong Bir(X_{min})$$ where $X_{min}$ is the minimal model of projective variety $X$
For any surface of non-negative Kodaira dimension, we have
$$Aut(X) ⊂ Bir(X) \cong Bir(X_{min}) \cong Aut(X_{min})$$
If $X$ be a Fano variety of $dim\geq 5$ and Picard group is generated by anticanonical divisor of the variety $X$, then it is conjecture that $Aut(X)\cong Bir(X)$
Moreover for moveable log pair $(X,M_X)$ which is log Calabi-Yau pair, with at worst canonical singularities then $Aut(X)\cong Bir(X)$