國立清華大學 數學系

Abstract：
The geometry of a smooth projective algebraic variety $X$ is often best understood through the numerical properties of its (pluri)canonical bundle(s) $omega_X^m$. As algebraic varieties naturally come in families, it is natural to ask how sections of $omega_X^m$ behave under deformations. A famous theorem by Siu states that, if $varphicolon Xto S$ is a smooth projective family of algebraic varieties, every global pluricanonical form on $X_s$ can be extended to $X$. This theorem and the ideas involved in its proof have been extremely influential in higher-dimensional algebraic geometry: they play a key role in the construction of moduli for varieties of general type, as well as in the construction of flips in arbitrary dimension. However, the proof of Siu's theorem is analytic and as such it does not extend to positive and mixed characteristic. In this talk I will discuss how one can generalize Siu's theorem to this latter setting, discuss some related pathological behaviors and the implications these have for moduli theory and the Minimal Model Program.