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Haidong Liu

Publications and source records attributed to Haidong Liu.

At least 19 recordsLinked to original sources
arXiv2026

Compactness of positive radial Solution Sets and corresponding $L^2$-Mass sets for "Zero Mass" Quasi-linear Schr\"odinger Equations

We study the compactness of two sets of positive radial solutions to ``zero mass'' quasi-linear Schr\"odinger equation \[ -\Delta u-u\Delta(|u|^2)=g(u) \quad\text{in }\mathbb R^N,\quad N\ge5. \] For nonlinearities that are either strictly subcritical or asymptotically critical at infinity, we prove that the set of least energy positive radial solutions is nonempty and compact in the natural space. Moreover, in the strictly subcritical case and under additional assumptions, we show that the set of all finite $L^2$-mass positive radial solutions is compact in \(L^2(\mathbb R^N)\). The proof relies on uniform decay estimates for the corresponding positive radial solutions of the transformed semilinear equation, which yield the required uniform \(L^2\)-tail control.

math.AP↗
arXiv2026

Kawamata--Miyaoka type inequality for varieties with nef anticanonical divisor

Let $X$ be an $\epsilon$-lc projective variety of dimension $n\geq 2$ such that $-K_X$ is nef and $0<\epsilon \leq 1$ is a real number. Then for any nef divisors $D_1,\dots,D_{n-2}$, \[ c_1(X)^{2}\cdot D_1\cdots D_{n-2}\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot D_1\cdots D_{n-2}. \] In particular, there exists a Kawamata--Miyaoka type inequality \[ c_1(X)^n\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot c_1(X)^{n-2}. \]

math.AG↗
arXiv2026

On Fano indices of weighted projective spaces

The Sylvester sequence is defined recursively by $s_1=2$ and $s_i=s_{1}\cdots s_{i-1}+1$. In this paper, we prove that the Fano index of an $n$-dimensional well-formed weighted projective space with canonical singularities is bounded above by \[ (s_n-1)(2s_n-3). \] This gives an affirmative answer to a conjecture of Chengxi Wang for weighted projective spaces and $\mathbb Q$-factorial toric Fano varieties with Picard number one. We also investigate the distribution of Fano indices among $4$-dimensional weighted projective spaces. As the distribution of Fano indices of weighted projective spaces coincides with that of indices of terminal Calabi--Yau varieties in dimension $n\leq 3$, we expect this coincidence to persist also in dimension 4, and more generally, in all dimensions.

math.AG↗
arXiv2025

A Kawamata--Miyaoka type inequality for Fano varieties of arbitrary Picard number

Let $X$ be a $\mathbb Q$-factorial canonical weak Fano variety of dimension $n\geq 2$. We show that if the $\mathbb Q$-Fano index $q_{\mathbb Q}(X)\geq 3$, then $X$ satisfies a Kawamata--Miyaoka type inequality: \[c_1(X)^n\leq 4\,\hat c_2(X)\cdot c_1(X)^{n-2}.\] As an application, we show that the $\mathbb Q$-Fano index of a Gorenstein canonical Fano $3$-fold lies in the set $\{m\in\mathbb Z_{>0}\mid m\leq 22\} \cup\{24,30,42\}$.

math.AG↗
arXiv2025

A canonical Fano threefold has Fano index $\leq 66$

We show that the $\mathbb{Q}$-Fano index of a canonical weak Fano $3$-fold is at most $66$. This upper bound is optimal and gives an affirmative answer to a conjecture of Chengxi Wang in dimension $3$. During the proof, we establish a new Riemmann--Roch formula for canonical $3$-folds and provide a detailed study of non-isolated singularities on canonical Fano $3$-folds, concerning both their local and global properties. Our proof also involves a Kawamata--Miyaoka type inequality and geometry of foliations of rank $2$ on canonical Fano $3$-folds.

math.AG↗
arXiv2023

Kawamata--Miyaoka type inequality for $\mathbb{Q}$-Fano varieties with canonical singularities

Let $X$ be an $n$-dimensional normal $\mathbb{Q}$-factorial projective variety with canonical singularities and Picard number one such that $X$ is smooth in codimension two, $-K_X$ is ample and $n\geq 2$. We prove that $X$ satisfies the following Kawamata--Miyaoka type inequality: \[ c_1(X)^n< 4 c_2(X)\cdot c_1(X)^{n-2}. \] If additionally $X$ is a threefold with terminal singularities, then a stronger inequality is also obtained.

math.AG↗
arXiv2023

On the log version of Serrano's conjecture

In this paper, we continue the study of Serrano's conjecture in low dimensions. We focus on two special cases of the log version of Serrano's conjecture: the ampleness conjecture and the log version of Campana--Peternell's conjecture. In dimension 3, we prove that the ampleness conjecture holds for non-canonical singularities; by the same method, we also prove that the log canonical version of Campana--Peternell's conjecture holds in dimension 3. In dimension 4, we improve the results on Campana--Peternell's conjecture by excluding the case that the numerical dimension of the anti-canonical divisor is 3. Specifically, we show that for a projective smooth fourfold $X$, if $-K_X$ is strictly nef but not ample, then $κ(X, -K_X)=0$ and $ν(X, -K_X)=2$; in this case, if we further assume that $X$ admits a Fano contraction $X\to Y$ onto a surface $Y$ induced by some extremal ray, then $ρ(X)=2$.

math.AG↗
arXiv2023

On a numerical criterion for Fano fourfolds

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano fourfold. To this aim, we completely solve the non-vanishing conjecture for strictly nef anti-canonical divisors in dimension 4.

math.AG↗
arXiv2023

Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors

In this paper, we study the Miyaoka type inequality on Chern classes of terminal projective $3$-folds with nef anti-canonical divisors. Let $X$ be a terminal projective $3$-fold such that $-K_X$ is nef. We show that if $c_1(X)\cdot c_2(X)\neq 0$, then $c_1(X)\cdot c_2(X)\geq \frac{1}{252}$; if further $X$ is not rationally connected, then $c_1(X)\cdot c_2(X)\geq \frac{4}{5}$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of $c_1(X)^{\dim X-2}\cdot c_2(X)$ for terminal weak Fano varieties and prove a Miyaoka--Kawamata type inequality for terminal weak Fano $3$-folds.

math.AG↗
arXiv2021

On the existence of rational curves on projective hyperkähler fourfolds

We show that if $X$ is a projective hyperkähler fourfold and there exists a nonzero effective divisor $D$ which is not of bi-elliptic type and contained in the boundary of the nef cone of $X$, then $X$ contains a rational curve. This is a very special case of Oguiso's conjecture for projective hyperkähler fourfolds.

math.AG↗
arXiv2021

Strictly nef divisors on K-trivial fourfolds

In this paper, we prove the ampleness conjecture and Serrano's conjecture for strictly nef divisors on K-trivial fourfolds. Specifically, we show that any strictly nef divisors on projective fourfolds with trivial canonical bundle and vanishing irregularity are ample.

math.AG↗
arXiv2020

Rational curves and strictly nef divisors on Calabi--Yau threefolds

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $ν(D)\neq 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.

math.AG↗
arXiv2020

Remarks on very basic slc-trivial fibrations

We study very basic slc-trivial fibrations. We show that restricting on any lc center of a very basic slc-trivial fibration, its moduli part is numerically trivial if and only if it is $\mathbb Q$-linearly trivial. We then prove that abundance conjecture for very basic slc-trivial fibrations holds true in dimension two when the moduli part is $\mathbb Q$-Cartier. As an application, we prove that the log canonical ring of a projective plt pair with Kodaira dimension 3 is finitely generated.

math.AG↗
arXiv2020

High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents

We study the coupled Hartree system $$ \left\{\begin{array}{ll} -Δu+ V_1(x)u =α_1\big(|x|^{-4}\ast u^{2}\big)u+β\big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -Δv+ V_2(x)v =α_2\big(|x|^{-4}\ast v^{2}\big)v +β\big(|x|^{-4}\ast u^{2}\big)v &\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$, $β>\max\{α_1,α_2\}\geq\min\{α_1,α_2\}>0$, and $V_1,\,V_2\in L^{N/2}(\mathbb{R}^N)\cap L_{\text{loc}}^{\infty}(\mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(\mathbb{R}^N)}+|V_2|_{L^{N/2}(\mathbb{R}^N)}>0$ is suitably small.

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