题目：Positivity of direct images and applications to birational geometry
报告人： 王隽永 博士 （中科院数学研究所）
报告时间地点： 4月6日上午10:00-11:50， 后主楼1124；4月8日、15日上午10:00-11:50，后主楼1220.
摘要：Let f :X -> Y be a fibre space between complex manifolds and let (L,h) be a Hermitian line bundle equipped with a singular Hermitian metric whose curvature current is semipositive. If locally over Y the manifold X is Kähler, then the L2 metric on the direct image sheaf f_*(K_{X/Y}+L) is a semipositive singular Hermitian metric in the sense of Griffiths (Deng-Wang-Zhang-Zhou’18). This can be seen as a generalization of Griffiths’s famous result on the variation of Hodge structures (where he treats the case : f smooth, i.e. f is a family of Kähler manifolds and L=O_X) and of Viehweg’s weak positvity theorem. In this talk I will present some applications of this positivity result to the birational geometry, as I briefly recall as following : the positivity theorem of direct images is powerful tool to study the canonical fibrations associated to varieties. For varieties with semipositive canonical bundle, this can be used to prove some special cases of the Iitaka C_{n,m} conjecture ; while for the uniruled case, it helps to show that the Albanese map or the maximal rationally connected fibration is locally trivial.
