arXiv subjects
Publications and source records attributed to Yinji Li.
We prove the uniqueness conjecture on the twisted Kähler-Einstein currents in big cohomology classes, by solving a problem of Bo Berndtsson posed in his work on the uniqueness theorem of Käher-Einstein metrics.
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.
Given a cohomology $(1,1)$-class $\{β\}$ of compact Hermitian manifold $(X,ω)$ possessing a bounded potential and fixed a model potential $ϕ$, motivated by Darvas-Di Nezza-Lu and Li-Wang-Zhou's work, we show that degenerate complex Monge-Ampère equation $(β+dd^c φ)^n=e^{λφ}μ$ has a unique solution in the relative full mass class $\mathcal{E}(X,β,ϕ)$, where $μ$ is a non-pluripolar measure on $X$ and $λ\geq0$ is a fixed constant. As an application, we give an explicit description of Lelong numbers of elements in $\mathcal{E}(X,β,ϕ)$ which generalized a theorem of Darvas-Di Nezza-Lu in the Hermitian context.
Let $(X,ω)$ be a compact Hermitian manifold of complex dimension $n$. Let $β$ be a smooth real closed $(1,1)$ form such that there exists a function $ρ\in \mbox{PSH}(X,β)\cap L^{\infty}(X)$. We study the range of the complex non-pluripolar Monge-Ampère operator $\langle(β+dd^c\cdot)^n\rangle$ on weighted Monge-Ampère energy classes on $X$. In particular, when $ρ$ is assumed to be continuous, we give a complete characterization of the range of the complex Monge-Ampère operator on the class $\mathcal E(X,β)$, which is the class of all $φ\in \mbox{PSH}(X,β)$ with full Monge-Ampère mass, i.e. $\int_X\langle (β+dd^cφ)^n\rangle=\int_Xβ^n$.
Let $(X,ω)$ be an $n$-dimensional compact Hermitian manifold with $ω$ a pluriclosed Hermitian metric, i.e. $dd^cω=0$. Let $\{α\},\{β\}\in H^{1,1}_{BC}(X,\mathbb R)$ be two nef classes, such that $α^n-nα^{n-1}\cdotβ>0$. In this short note, we prove that if there is a bounded quasi-plurisubharmonic potential $ρ$, such that $α+dd^cρ\geq 0$ in the weak sense of currents, then the class $\{α-β\}$ contains a Kähler current. This gives a partial solution of Demailly's transcendental Morse inequalities conjecture.
In this paper, we first answer Chen-Zhang's problem on $p$-Bergman metric proposed in \cite{CZ22}. Second, we prove the off-diagonal p-Bergman kernel function $K_p(z,w)$ is Hölder continuous of order (1-$\varepsilon$) about the second component when $p>1$ for any $\varepsilon>0$, which improves the corresponding result of Chen-Zhang. Moreover, we prove the asymptotic behavior of the maximizer of $p$-Bergman kernel as $p\rightarrow 1^-$. Finally, we give a characterization of a class of holomorphic functions on $\mathbb{B}^1$ to be $L^p$-integrable.
We show the existence and uniqueness of bounded solutions to the degenerate complex Monge-Ampère type equations on compact Hermitian manifolds. We also study the asymptotics of these solutions. As applications, we give partial answers to the Tosatti-Weinkove conjecture and Demailly-P\u aun conjecture.