Kawamata-Viehweg and Lipman's vanishings for sequences of blow-ups at regular closed points

Abstract

The birational Kawamata-Viehweg vanishing theorem (KVV for short) is known to hold in considerable generality in equal characteristic zero, but fails in general in positive characteristic. We study a vanishing conjecture of Lipman for sequences of blow-ups at closed points of the spectrum of a regular local ring. We prove the conjecture when the ring has dimension at most three. We also establish a relative Kawamata-Viehweg vanishing theorem for exceptional nef Q-divisors when all blow-up centers are rational over the residue field and the positive residue characteristic is at least the dimension of the ring minus one. Both results allow mixed characteristic.