arXiv subjects
Publications and source records attributed to Chi-Kang Chang.
We obtain inequalities for isolated $cD_n, cE_6, cE_7$ and $cE_8$ singularities, analogues of the $4\mu^2/(n+1)$-inequality for isolated $cA_n$ singularity in \cite{KOP}. These inequalities are sharp. This enables us to prove the birational rigidity of certain families of Fano $3$-fold weighted hypersurfaces, which contain only terminal quotient singularities and one $cD_n$ singularity.
Given a fibration $f: X \to Y$ with normal general fibre $X_y$, over a field of any characteristic, we establish the Iitaka-type inequality $κ(X,-K_X) \leq κ(X_y,-K_{X_y})+κ(Y,-K_Y)$ whenever the $\mathbb{Q}$-linear series $|-K_X|_{\mathbb{Q}}$ has good singularities on $X_y$.
In this article, we will prove the Generalized Nonvanishing Conjecture holds for threefolds with either $κ>0$ or $q>0$. As a result, we can prove the Iitaka conjecture $C_{n,m}$ holds for $n=7$ if the source space has non-negative Kodaira dimension, if the general fibre has positive Kodaira dimension, or if the base space is not threefold with $κ=q=0$. In particular, $C^-_{n,m}$ holds if $n\leq 7$.
Let $f:X\rightarrow Y$ be an algebraic fibre space between normal projective varieties and $F$ be a general fibre of $f$. We prove an Iitaka-type inequality $κ(X,-K_X)\leq κ(F,-K_F)+κ(Y,-K_Y)$ under some mild conditions. We also obtain some more results relates the positivity of $-K_X$ and $-K_Y$.