threefold toric
THEOREM 1.2. Let (X, D = ∑ diDi ) be a three-dimensional projective variety over C such that KX + D ≡ 0 and (X, D) has only purely log terminal singularities. Then (1.3) ∑ di ≤ rank(Weil(X)/≈) + 3 . Moreover, if the equality holds, then up to isomorphisms one of the following holds: (i) X P3, D = 0 or D = P2; (ii) X P1 × P2, D = 0 or D = {pt} × P2 or D = P1 × {line} (iii) X P(OP1 ⊕ OP1 ⊕ OP1 (d)), d ≥ 1, D is the section corresponding to the surjection OP1 ⊕ OP1 ⊕ OP1 (d) → OP1 (d); (iv) X P1×P1×P1, D = 0 or D = {pt}×P1×P1 or D = {pt1, pt2}×P1×P1; (v) X P(OP2 ⊕ OP2(d)), d ≥ 1, D is the negative section, or a disjoint union of two sections, one of them is negative; (vi) X P(OP1×P1 ⊕ L), L ∈ Pic(P1 × P1), D is the negative section, or a disjoint union of two sections, one of them is negative. In all cases (X, D ) is toric.