Q~ +I De Rham complex ~iX,Y) a sends toroidal ~X,Y) of the pair pair.

1). By R i e m a r m - R o c h , A - F i - F j i s effective if i * j . Thus (A-Fi)°F = (A-Fi-Fj)°F+F j °F _>FfF. If F°Fj > 1 for some j¢ i we are done. F = 9+F°F i, and (A-Fi)°F = 3 - 3F°Fi . F i = 3, AoF = 4. (F+Fi) 2 -(A-(F+Fi))-) = 6 0 - 4 9 > 0. The latter contradicts the Hodge Index theorem. L e m m a 3. A <_5. Then D e <-O. PROOF. By Hodge's Index theorem: A2D 2 = 10D 2 -< (A,D) ~ _<25, that yields D 2 _<2. A s s u m e D 2 = 2. Let IDI = IMI+A, where A is the fixed part of IDt. zX _<5, A = I3 or A-M < 4.