arXiv subjects
Publications and source records attributed to Qunhuan Liu.
We show that the Grauert blow-down of a holomorphic negative line bundle $L$ over a compact complex space is a quadratic transform if and only if $k_0L^*$ is very ample and $(k_0+1)L^*$ is globally generated, where $k_0$ is the initial order of $L^*$, namely, the minimal integer such that $k_0^*$ has nontrivial holomorphic section.
For plurisubharmonic functions $φ$ and $ψ$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $φ$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max\{φ(z), ψ(Az)\})^n$ on $\mathbb{C}^n$ is zero for $A\in \mbox{Hom}(\mathbb{C}^n,\mathbb{C}^m)=\mathbb{C}^{nm}$ outside a pluripolar set. For a plurisubharmonic function $φ$ near the origin in $\mathbb{C}^n$, we introduce a new concept coined the log truncated threshold of $φ$ at $0$ which reflects a singular property of $φ$ via a log function near the origin (denoted by $lt(φ,0)$) and derive an optimal estimate of the residual Monge-Ampère mass of $φ$ at $0$ in terms of its higher order Lelong numbers $ν_j(φ)$ at $0$ for $1\leq j\leq n-1$, in the case that $lt(φ,0)<\infty$. These results provide a new approach to the zero mass conjecture of Guedj and Rashkovskii, and unify and strengthen well-known results about this conjecture.
We prove that the irreducible desingularization of a singularity given by the Grauert blow down of a negative holomorphic vector bundle over a compact complex manifold is unique up to isomorphism, and as an application, we show that two negative line bundles over compact complex manifolds are isomorphic if and only if their Grauert blow downs have isomorphic germs near the singularities. We also show that there is a unique way to modify a submanifold of a complex manifold to a hypersurface, namely, the blow up of the ambient manifold along the submanifold.
We first present the mixed Hilbert-Samuel multiplicities of analytic local rings over \mathbb{C} as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give estimates or an exact formula for the multiplicities of isolated singularities that given by the Grauert blow-downs of negative holomorphic vector bundles.