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Register a new userOn the pluricanonical maps of varieties of intermediate Kodaira dimension
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Authors
Xiaodong Jiang
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In this paper we will prove a uniformity result for the Iitaka fibration $f:X rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $kappa(X)=rm{dim}(X)-1$.
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In this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group $G$. It is shown that only for $G = operatorname{He(3)}, mathbb Z_3^2$, and only for dimension $geq 4$ such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension $4$ is given. For the other finite groups a strong structure theorem for rigid quotients is proven.
John Lesieutre constructed an example satisfying $kappa_sigma e kappa_ u$. This says that the proof of the inequalities in Theorems 1.3, 1.9, and Remark 3.8 in [O. Fujino, On subadditivity of the logarithmic Kodaira dimension, J. Math. Soc. Japan 69 (2017), no. 4, 1565--1581] is insufficient. We claim that some weaker inequalities still hold true and they are sufficient for various applications.
In the present paper, we study the (twisted) 3-canonical map of varieties of Albanese fiber dimension one. Based on a theorem about the regularity of direct image of canonical sheaves, we prove that the 3-canonical map is generically birational when the genus of a general fiber of the Albanese map is 2.
For an irregular variety $X$ of general type, we show that if a general fiber $F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic condition, the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ . We then show that the condition that the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ can often be applied to prove the birationality of certain pluricanonical maps of $X$.
We prove that the log canonical ring of a projective log canonical pair in Kodaira dimension two is finitely generated.
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