My important query is the next:

Let $f: mathbb{CP}^n to mathbb{CP}^n$ be a holomorphic endomorphism of diploma $d ge 2$ of $mathbb{CP}^n$ .

1.Let $X subset mathbb{CP}^n$ be an irreducible algebraic set such that $f(X) = X$. Set $g:= f|_X$. Denote by $V subset X$ the minimal (with respect the inclusion) algebraic set such that

$$g: X setminus g^{-1}(V) to X setminus V$$

is a protecting, i.e. $V$ is the ramification locus of $g$. Is $V$ at all times an algebraic set of codimension one in $X$ ?

Two partial questions, whose reply are equally to me as the primary query.

2.Do there exist an irreducible algebraic set $X$ such that $f: X to X$ and an algebraic set $V subset X$ of codimension no less than $2$ in $X$ such that $f: X setminus f|_X^{-1}(V) to X setminus V$ is a protecting ?

3.Does there exists an irreducible algebraic set $X subset mathbb{CP}^n$ such that $f(X) = X$ and $f: X to X$ is a neighborhood biholomorphism ?

### Motivation:

My motivation for this query is the category of *post-critically finite endomorphisms of $mathbb{CP}^n$.* Extra exactly, let $f : mathbb{CP}^n to mathbb{CP}^n$ be a holomorphic endomorphism. The essential worth set $V_f$ of $f$ is an algebraic set such that

$$f: mathbb{CP}^n setminus f^{-1}(V_f) to mathbb{CP}^n setminus V_f$$

is a protecting. The map $f$ known as *post-critically finite* if $$PC(f):=bigcuplimits_{j ge 0} f^{circ j}(V_f)$$ is an algebraic set of codimension one among $mathbb{CP}^n$, the place $f^{circ m}:= f circ ldots circ f$ is the $n$-th iterate of $f$. In different phrases, for each irreducible part $Gamma$ of $PC(f)$, $f^{circ ok}(Gamma)= f^{circ (ok+m)}(Gamma)$ for some $ok ge 0, mge 1$, i.e. $f^{circ ok}(Gamma)$ is invariant by $f^{circ m}$. I wish to give the identical notion of being *post-critically finite* for the restriction $f^{circ m}$ to $X:=f^{circ ok}(Gamma)$. Set $g:= f^{circ m}|_{X}$.

So step one is to outline what’s the essential worth set $V_g$ of $g$. Naturally, we are able to select a set $V_g$ such that

$$g: X setminus g^{-1}(V_g) to X setminus V_g$$

is a protecting. Since $g$ is the restriction of $f^{circ m}$, one candidate can merely be $X cap V_{f^{circ m}}$. Nevertheless, $V_{f^{circ m}}$ and $X$ have primarily no relation, the intersection will be very wild. For instance, when $X subset V_{f^{circ m}}$ or when it will probably have irreducible part of a number of dimension. I wish to get a codimension set in $X$. Nevertheless, at this level, I wonder if can certainly discover the case of codimension increased than one

**2.** Do there exist an irreducible algebraic set $X$ such that $f: X to X$ and an algebraic set $V subset X$ of codimension no less than $2$ in $X$ such that $f: X setminus f|_X^{-1}(V) to X setminus V$ is a protecting ?

One other selection of definition is that, the essential worth set is the picture of the essential set. Exactly, the essential set $C_f$ is the set of factors the place the by-product of $f$ isn’t invertible. Then, $V_f = f(C_f)$. The truth that $C_f$ is an algebraic set of codimension one among $mathbb{CP}^n$ is non trivial. The reason being that, $C_f$ is the locus of vanishing the Jacobian determinant of $f$. Moreover, the Jacobian determinant of an endomorphism of diploma $d$ is a polynomial of diploma $(n+1)(d-1)$, because of the computation of the classical endomorphism $[x_1:ldots:x_n] mapsto [x_1^d:ldots:x_n^d]$ and the truth that holomorphic endomorphisms kind an open linked set within the area of meromorphic endomorphisms of $mathbb{CP}^n$. When $X$ is a easy, the essential set $C_g$ of $g$ will be outlined in the identical style, i.e. the vanishing of the Jacobian determinant. Within the paper DYNAMICS OF POST-CRITICALLY FINITE MAPS IN HIGHER DIMENSION, Mathieu Astorg, 2018, the creator confirmed that $C_g$ is included within the intersection of $X$ and irreducible elements of $C_f$ apart from $X$. When $X$ is singular, we are able to nonetheless outline the by-product map on the Zariski tangent area, and we are able to nonetheless outline the essential locus is the set of factors the place the by-product isn’t invertible. Can or not it’s empty ? This results in the third query.

**3.** Does there exists an irreducible algebraic set $X subset mathbb{CP}^n$ such that $f(X) = X$ and $f: X to X$ is a neighborhood biholomorphism ?

It might be a simple query. Any remark, and particularly, reference recommendations about holomorphic maps on algebraic/analytic varieties are welcomed.