complex dynamic system and affine algebraic variety

Question

Let $M^n$ be a $n$-dimensional noncompact complex manifold. In "The density property for

complex manifolds and geometric structures II", Dror Varolin showed that some open set of

$M$ is biholomorphic to $C^n$ if there is a biholomorphsm $F:M \rightarrow M$ satisfying

certain density property.

More precisely, he proved the following theorem:

Theorem: Let $F$ be a biholomorphism from a complex manifold $M$ to itself and let $p$ be a

fixed point of $F$. Fix a complete Riemannian metric $g$ on $M$ and define

$$U=:\{{x \in M: lim_{k\rightarrow \infty} d_g (F^k(x), p)=0}\},$$

where $d_g$ is the Riemannian distance of $g$. Then $U$ is biholomorphic to $C^n$ provided

that $U$ contains an open neighborhood of $p$. In particular, if $U=M$, then $M$ is biholomorphic to $C^n$.

Now let $M$ be a noncompact complex manifold and $F$ be a biholomorphism on $M$.

Under what kind of conditions about $F$, we can show that $M$ is biholomorphic to

an affine algebraic variety or quasi-projective variety?

Answer 1

Your question correspond to Demailly's analytical characterization of Affine varieties

Let $X$ be a complex analytic manifold of dimension $n$. Then $X$ is biholomorphically equivalent to the affine algebraic manifold $X_{alg}$ if and only if the cohomology spaces $H^{2q}(X,\mathbb R)$ have finite dimension and there exists a strictly plurisubharmonic exhaustion function $\phi$ of class $C^\infty$ on $X$ such that $$vol(X)=∫_X(dd^c\phi)^n<+\infty$$ and the Ricci curvature of the metric $β=dd^c(e^\phi)$ has the following estimate

$$Ric(β)≥−\frac{1}{2}dd^c\psi$$ where $ψ∈C^0(X,\mathbb R),\psi≤A\phi+B $

See my answer also