Examples/classification of algebraic symplectomorphisms I'm curious about examples of algebraic automorphisms of complex varieties which are symplectomorphism. For instance, can we classify the algebraic symplectomorphisms of $\mathbb{P}_{\mathbb{C}}^n$ with regard to the Fubini-Study form? Or can we give large classes thereof for $\mathbb{P}_{\mathbb{C}}^n$ or other complex algebraic varieties? If understand a comment on this question correctly, there ought to be an ample supply....

A general remark about Kahler manifolds is that a biholomorphic automorphism is symplectomorphic if and only if it is an isometry. (Generally, if a diffeomorphism preserves two out of three structures given by a Kahler metric: hermitian, symplectic, complex, then it preserves all three.) In the case of $CP^n$ with the FS-metric, the group of biholomorphic isometries is easily seen to be equal to $PU(n+1)$. As for more general examples, you can consider  biholomorphic isometries of compact hermitian symmetric spaces (these spaces will be algebraic). Helgason's book
Helgason, Sigurdur, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80. New York-San Francisco-London: Academic Press. XV, 628 p.  (1978). ZBL0451.53038.
contains a description of these groups (I do not remember the details). For general smooth algebraic varieties, describing the groups of biholomorphic isometries will be difficult. On general grounds, all you can say that this group will be a compact Lie group.
