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SUMMARY:Takumi Murayama (Princeton University)
DTSTART:20220111T170000Z
DTEND:20220111T180000Z
DTSTAMP:20260423T040003Z
UID:ZAG/176
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ZAG/176/">Th
 e Kawamata–Viehweg vanishing theorem for schemes</a>\nby Takumi Murayama
  (Princeton University) as part of ZAG (Zoom Algebraic Geometry) seminar\n
 \n\nAbstract\nIn 1953\, Kodaira proved what is now called the Kodaira vani
 shing theorem\, which states that if L is an ample divisor on a complex pr
 ojective manifold X\, then H^i(X\,-L) = 0 for all i < dim(X). Since then\,
  Kodaira's theorem and its generalizations for complex projective varietie
 s – in particular\, the Kawamata–Viehweg vanishing theorem and its rel
 ative version due to Kawamata–Matsuda–Matsuki – have become indispen
 sable tools in algebraic geometry over fields of characteristic zero\, in 
 particular in birational geometry and the minimal model program. However\,
  while the goal in the minimal model program is to study birational equiva
 lences between projective varieties\, recent progress in the minimal model
  program has shown that it would be very useful to have a version of the K
 awamata–Viehweg vanishing theorem that holds for schemes that are not ne
 cessarily projective varieties. Recently\, I proved the relative Kawamata
 –Viehweg vanishing theorem for schemes of equal characteristic zero. My 
 results are optimal given known counterexamples to vanishing theorems in p
 ositive and mixed characteristic\, and have many applications to both alge
 braic geometry and commutative algebra. In this talk\, I will discuss my v
 anishing theorem and several of its applications.\n
LOCATION:https://researchseminars.org/talk/ZAG/176/
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