arXiv subjects
Publications and source records attributed to Jheng-Jie Chen.
We obtain inequalities for isolated $cD_n, cE_6, cE_7$ and $cE_8$ singularities, analogues of the $4μ^2/(n+1)$-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.
In this paper, we complete the computation of the global log canonical thresholds of Burniat surfaces. Additionally, we determine the global log canonical thresholds for certain infinite families of surfaces of general type, constructed as $ \mathbb{Z}_2^n $-covers of $\mathbb{P}^2$ or $\mathbb{P}^1 \times \mathbb{P}^1$.
We prove that each divisorial contraction to a curve between terminal threefolds is a weighted blow-up under a suitable embedding. Moreover, we give a classification of the weighted blow-ups assuming that the curve is smooth.
We show that the set $\mathcal{T}_{3, \mathrm{sm}}^{\mathrm{can}}$ of smooth threefold canonical thresholds coincides with $\mathcal{T}_{2, \mathrm{sm}}^{\mathrm{lc}}=\mathcal{HT}_{2}$, where $\mathcal{HT}_{2}$ is the $2$-dimensional hypersurface log canonical thresholds characterized by Kuwata \cite{K99a, K99b}. We classify the set $\mathcal{T}_{3}^{\mathrm{can}}$ of threefold canonical thresholds. More precisely, we prove $\mathcal{T}_{3}^{\mathrm{can}}= \{0\} \cup \{\frac{4}{5}\} \cup \mathcal{T}_{3, \mathrm{sm}}^{\mathrm{can}}$.
In the paper we compute the global log canonical thresholds of the secondary Burniat surfaces with $K^2 = 5$. Furthermore, we establish optimal lower bounds for the log canonical thresholds of members in pluricanonical sublinear systems of the secondary Burniat surfaces with $K^2 = 5$.
We show that the set of threefold canonical thresholds satisfies the ascending chain condition. Moreover, we derive that threefold canonical thresholds in the interval $ (\frac{1}{2}, 1)$ consists of $ \{ \frac{1}{2}+\frac{1}{n}\}_{n \ge 3} \cup \{ \frac{4}{5}\}$.
We obtain that the nonzero accumulation points of the set of 3-fold canonical thresholds $\ct(X,S)$ are precisely $1/k$ where $k\ge 2$ is an integer and $S$ is an effective integral divisor of a projective 3-fold $X$ with only terminal singularities. Moreover, we generalize the ascending chain condition for the set of 3-fold canonical thresholds to pair.
Let $X$ be a complex smooth projective threefold of general type. Assume $q(X)>0$. We show that the $m$-canonical map of $X$ is birational for all $m\geq 5$.
We prove that if $X---> X^+$ is a threefold terminal flip, then $c_1(X).c_2(X)\leq c_1(X^+).c_2(X^+)$ where $c_1(X)$ and $c_2(X)$ denote the Chern classes. This gives the affirmative answer to a Question by Xie \cite{Xie2}. We obtain the similar but weaker result in the case of divisorial contraction to curves.
We prove that the general elephant $E_{X^+}\in |-K_{X^+}|$ is normal where $X--> X^+$ is a 3-fold terminal flip. Hence $E_{X^+}$ has at worst Du Val singularities. As a corollary, there exists no non-Gorenstein singularity of type $cE/2$, $cD/3$, nor $cAx/4$ on the flipped curve $C^+$.
We prove that there exist only finitely many families of Calabi-Yau quasismooth weighted complete intersections with every fixed dimension $m$. This generalizes a result of Johnson and Kollár to higher codimensions.
We prove two conjectures on weighted complete intersections and give the complete classification of threefold weighted complete intersections in weighted projective space that are canonically or anticanonically embedded.