This chapter is devoted to the basic vanishing theorems for integral divisors. The prototype is Kodaira’s result that if A is an ample divisor on a smooth complex projective variety X, then O X (K X + A) has vanishing higher cohomology. An important extension, due to Kawamata and Viehweg, asserts that the statement remains true assuming only that A is nef and big. These and related vanishing theorems have a vast number of applications to questions central to the focus of this book.
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