Abstract. We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if so is the log canonical divisor of its normalization. We also treat many other applications.

We consider a canonical bundle formula for generically finite proper surjective morphisms and obtain subadjunction formulae for minimal log canonical centers of log canonical pairs. We also treat related topics and applications.

We show that every Mori dream space of globally Fregular type is of Fano type. As an application, we give a characterization of varieties of Fano type in terms of the singularities of their Cox rings.

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We prove the W O-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic p > 5. As a consequence, any klt Fano threefold over a finite field has a rational point.