in the classification theory of higher dimensional algebraic varieties a central role is played by the canonical divisor $k_x$ and its multiples. a famous theorem of y. t. siu states that if ${\pi:x\longrightarrow}$ t is a smooth projective family of varieties, then the plurigenera of the fibres $\lbrace h^0(x_t, mk_{x_t})\rbrace_{m\geq 0}$ are constant in t. despite being an algebraic problem, siu's proof employs methods which are essentially analytic in nature. after giving an overview of the techniques involved, we outline a path to a possible algebraic proof, based on a reduction to the general type case via the iitaka fibration.