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SUMMARY:Minimal slopes and singular solutions for complex Hessian equation
 s - Ved Datar (Indian Institute of Science)
DTSTART:20240515T091500Z
DTEND:20240515T101500Z
UID:TALK215647@talks.cam.ac.uk
DESCRIPTION:It is well known that solvability of the complex Monge- Ampere
  (CMA) equation on compact Kaehler manifolds is related to the positivity 
 of certain intersection numbers. In fact\, this follows from combining Yau
 &rsquo\;s resolution of the Calabi conjecture\, with Demailly and Paun&rsq
 uo\;s generalization of the classical Nakai-Mozhesoin criteria. This corre
 spondence was recently extended to a broad class of complex non-linear PDE
 s including the J-equation and the deformed Hermitian-Yang-Mills (dHYM) eq
 uations by the work of Gao Chen and others. A natural question to ask is w
 hether solutions (necessarily singular) exist in any reasonable sense if t
 he Nakai criteria fails. A motivating example is that of CMA equations in 
 big classes. In this talk\, I will provide an overview of a program for co
 nstructing such singular solutions for the J equation and the dHYM equatio
 ns\, and state some conjectures and problems. Analogous to the characteriz
 ation of volumes of big classes\, a key point is to define a minimal slope
  using birational models. Finally\, I will outline how to resolve some of 
 these questions on Kahler surfaces and some manifolds with large symmetry 
 groups. This is joint work with Ramesh Mete and Jian Song.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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