appendix 2

PROPOSITION 2.1   Let $d$ be a positive integer. Let $(X,\omega_X)$ , $(Y,\omega_Y)$ be smooth symplectic algebraic varieties of dimension $2 n$ over a field $k$ . Let $\phi:X\to Y$ be a symplectic morphism. That means, it is a morphism which preserves the symplectic structure :

$$\phi^*(\omega_Y)=\omega_X$$

Then the tangent map of $\phi$ is of full rank at every point $P$ on $X$ .

We make an effective use of the theory of the Pfaffian $\operatorname{Pfaff}(M)$ of a given matrix $M$ . A good reference is in [2, XV,§9]. Especially important theorem we need to know is the following lemma.

LEMMA 2.2 ([2, XV,Theorem 9.1])   Let $R$ be a commutative ring. Let $(m_{i j})=M$ be an alternating matrix with $g_{i j}\in R$ . Then

$$\det(M)=(\operatorname{Pfaff}(M))^2.$$

Furthermore, if $C$ is an $n\times n$ matrix in $R$ , then

$$\operatorname{Pfaff}(C M {}^t C)=\det(C) \operatorname{Pfaff}(M).$$

PROOF.. (of Proposition 2.1) We may assume that $k$ is algebraically closed and that $P$ is a $k$ -valued point. Let us represent the tangent map of $f$ at $P$ by $T_P f$ . One may choose a local coordinate system $x_1,\dots,x_{2n}$ on $X$ around $P$ such that the symplectic form $\omega_X$ at $P$ is represented by the matrix $h$ when expressed in terms of $d x_1,\dots,d x_{2n}$ . Likewise one may choose a local coordinate system $y_1,\dots,y_{2 n}$ around $f(P)$ such that the symplectic form $\omega_Y$ at $f(P)$ is represented by the matrix $h$ when expressed in terms of $d y_1,\dots,d y_{2n}$ . Then using the base $\partial/\partial x_1, \partial/\partial x_2 ,\dots \partial/\partial x_{2n}$ $\partial/\partial y_1, \partial/\partial y_2 ,\dots \partial/\partial y_{2n}$ , $T_P f$ may be identified with a $2 n \times 2 n$ -matrix. Since by hypothesis $T_P f$ reserves the symplectic form, we have

$$({}^t T_P f) h (T_P f)=h.$$

Let us compare the Pfaffian of the both hand sides.

$$\det(T_P f)^2 \cdot \operatorname{Pfaff}(h)=\operatorname{Pfaff}(({}^t T_P f) h T_P f)=\operatorname{Pfaff}(h).$$

Since $\operatorname{Pfaff}(h)=1$ , we conclude that the determinant of $T_P f$ should be equal to 1 or (-1). $\qedsymbol$

Note: It goes without saying that when $X$ is connected, and if the coordinate systems $x_1,\dots,x_{2n}$ and $y_1,\dots,y_{2 n}$ are able to chosen globally (for example if $X,Y$ are affine space $\mathbb{A}^{2 n}=\operatorname{Spec}k[T_1,T_2,T_3,\dots,T_{2 n}]$ with

$$\omega=d T_1 \wedge d T_{n+1} + d T_2 \wedge d T_{n+2} + d T_3 \wedge d T_{n+3} +\dots +d T_n \wedge d T_{2 n}$$

as the symplectic form), then the Jacobian of $f$ should either be $1$ on the whole of $X$ or be $-1$ on the whole of $X$ .