algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety moment map; polarization; reductive group; complex reductive group; algebraic group action; quotient morphism; ample line bundle; semistability; Stein holomorphic map; Kähler form; ample \(G\)-line bundle; semistable points; Geometric Invariant Theory 