In this dissertation, we study the birational geometry of log pairs over the field of complex numbers, with an emphasis on the positivity properties of log pairs and their applications. First, we study the nonvanishing conjecture in the minimal model program. We prove the nonvanishing conjecture for uniruled log canonical pairs of dimension $n$, assuming the nonvanishing conjecture for smooth projective varieties in dimension $n-1$. Second, we prove that pushforwards of klt pairs under morphisms to abelian varieties satisfy some positivity properties, which include global generation after pullback by an isogeny, the existence of the Chen--Jiang decompositions and so on. These are applied to some effective results for linear systems on irregular varieties. Third, we prove some vanishing and torsion-freeness results for higher direct images of adjoint pairs satisfying relative abundance and nefness conditions. We apply these to generic vanishing and weak positivity. Finally, we study some conjectures on the behavior of Kodaira dimension proposed by Popa. We prove an additivity result for the log Kodaira dimension of algebraic fiber spaces over abelian varieties, a superadditivity result for algebraic fiber spaces over varieties of maximal Albanese dimension, as well as a subadditivity result for log pairs over abelian varieties.

Creator

• Meng, Fanjun

DOI

• https://doi.org/10.21985/n2-c0t4-n753

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:15958
• etdadmin_upload_883432

Keyword

• Birational Geometry
• Positivity Properties
• Log Pairs

Date created

• 2022-01-01

Resource type

• Dissertation

Rights statement

• In Copyright