Jie Liu

Research interest

I am mainly working on complex algebraic geometry and complex geometry.

• My MathSciNet personal page: mathscinet.ams.org/mathscinet/author?authorId=1308679
• My zbMATH personal page: zbmath.org/authors/?q=ai%3Aliu.jie.6
• My arXiv personal page: arxiv.org/a/liu_j_10
• My ORCID personal page: orcid.org/0000-0001-8131-5909

Preprints

1. The spectral base and quotients of bounded symmetric domains by cocompact lattices (with Siqi He and Ngaiming Mok)
   [abstract]   [arXiv]   Preprint (2024)
   In this article, we explore Higgs bundles on a projective manifold \(X\), focusing on their spectral base —-- a concept introduced by T. Chen and B.Ngô. The spectral base is a specific closed subscheme within the space of symmetric differentials. We observe that if the spectral base vanishes, then any reductive representation \(\rho: \pi_1(X) \to \text{GL}_r(\mathbb{C})\) is both rigid and integral. Additionally, we prove that for \(X=\Omega/\Gamma\), a quotient of a bounded symmetric domain \(\Omega\) of rank at least \(2\) by a torsion-free cocompact irreducible lattice \(\Gamma\), the spectral base indeed vanishes, which generalizes a result of B. Klingler.
2. Kawamata--Miyaoka type inequality for canonical \(\mathbb{Q}\)-Fano varieties II: terminal \(\mathbb{Q}\)-Fano threefolds (with Haidong Liu)
   [abstract]   [arXiv]   Preprint (2024)
   We prove an optimal Kawamata--Miyaoka type inequality for terminal \(\mathbb{Q}\)-Fano threefolds with Fano index $\geq 3$. As an application, we obtain that any terminal \(\mathbb{Q}\)-Fano threefold \(X\) satisfies the following Kawamata--Miyaoka type inequality \(c_1(X)^3< 3 c_2(X)c_1(X)\).
3. On the spectral variety for rank two Higgs bundles (with Siqi He)
   [abstract]   [arXiv]   Preprint (2023)
   In this article, we study the Hitchin morphism over a smooth projective variety \(X\). The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which in general not surjective when the dimension of \(X\) is greater than one. Chen-Ngô introduced the spectral base, which is a closed subvariety of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen-Macaulayfications of the spectral varieties. For rank two Higgs bundles over a projective manifold \(X\), we explicitly construct a finite Cohen-Macaulayfication of the spectral variety as a double branched covering of \(X\), thereby confirming Chen-Ngô's conjecture in this case. Moreover, using this Cohen-Macaulayfication, we can construct the Hitchin section for rank two Higgs bundles, which allows us to study the rigidity problem of the character variety and also to explore a generalization of the Milnor-Wood type inequality.
4. The algebra of symmetric tensors on smooth projective varieties (with Arnaud Beauville)
   [abstract]   [arXiv]   Preprint (2023)
   We discuss in this note the \(\mathbb{C}\)-algebra \(H^0(X,\mathsf{S}^{\bullet}T_X)\) for a smooth complex projective variety \(X\). We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.
5. Kawamata--Miyaoka type inequality for canonical \(\mathbb{Q}\)-Fano varieties (with Haidong Liu)
   [abstract]   [arXiv]   Preprint (2023)
   A \(\mathbb Q\)-Fano variety is a normal \(\mathbb Q\)-factorial projective variety with Picard number one whose anti-canonical divisor is ample. In this paper, we prove a Kawamata--Miyaoka type inequality \(c_1(X)^n< 4 c_2(X)c_1(X)^{n-2}\) for any canonical $\mathbb Q$-Fano variety $X$ of dimension $n$ which is smooth in codimension two. We also prove a stronger inequality for terminal \(\mathbb{Q}\)-Fano threefolds.
6. Symmetric tensors on the intersection of two quadrics and Lagrangian fibration (with Arnaud Beauville, Antoine Etesse, Andreas Höring and Claire Voisin)
   [abstract]   [arXiv]   Preprint (2023)
   Let \(X\) be an \(n\)-dimensional (smooth) intersection of two quadrics, and let \(T^*X\) be its cotangent bundle. We show that the algebra of symmetric tensors on \(X\) is a polynomial algebra in \(n\) variables. The corresponding map \(T^*X\rightarrow \mathbb{C}^n\) is a Lagrangian fibration, which admits an explicit geommetric description; its general fiber is an open subset of an abelian variety, quotient of a hyperelliptic Jacobian by a \(2\)-torsion subgroup. In dimension \(3\), \(\Phi\) is the Hitchin fibration of the moduli space of rank \(2\) bundles with fixed determinant on a curve of genus \(2\).
7. Algebraic fibre spaces with strictly nef relative anti-log canonical divisor (with Wenhao Ou, Juanyong Wang, Xiaokui Yang and Guolei Zhong)
   [abstract]   [arXiv]   Preprint (2021)
   Let \((X,\Delta)\) be a projective klt pair, and \(f:X\rightarrow Y\) a fibration to a smooth projective variety \(Y\) with strictly nef relative anti-log canonical divisor \(-(K_{X/Y}+\Delta)\). We prove that \(f\) is a locally constant fibration with rationally connected fibres, and the base \(Y\) is a canonically polarized hyperbolic projective manifold. In particular, when \(Y\) is a single point, we establish that \(X\) is rationally connected. Moreover, when \(\text{dim}(X)=3\) and \(-(K_X+\Delta)\) is strictly nef, we prove that \(-(K_X+\Delta)\) is ample, which confirms the singular version of a conjecture of Campana-Peternell for threefolds.

Accepted Papers

Publications

8. Normalised tangent bundle, varieties with small codegree and pseudoeffective threshold (with Baohua Fu)
   [abstract]   [arXiv]   Journal of the Institute of Mathematics of Jussieu 23 (2024), no. 1, 149-206.
   We propose a conjectural list of Fano manifolds of Picard number \(1\) with pseudoeffective normalized tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and hence the pseudoeffective cones of the projectivized tangent bundles of rational homogeneous spaces of Picard number \(1\) are explicitly determined by studying the total dual VMRT and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number \(1\).
9. Projective manifolds whose tangent bundle contains a strictly nef subsheaf (with Wenhao Ou and Xiaokui Yang)
   [abstract]   [arXiv]   Journal of Algebraic Geometry 33 (2024), no. 1, 1-53.
   Suppose that \(X\) is a projective manifold whose tangent bundle \(T_X\) contains a locally free strictly nef subsheaf. We prove that \(X\) is isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group \(\pi_1(X)\) is virtually solvable, then \(X\) is isomorphic to a projective space.
10. On moment map and bigness of tangent bundles of \(G\)-varieties
    [abstract]   [arXiv]   Algebra & Number Theory 17 (2023), no. 8, 1501–1532.
    Let $G$ be a connected algebraic group and let \(X\) be a smooth projective \(G\)-variety. In this paper, we prove a sufficient criterion to determine the bigness of \(TX\) using the moment map \(\Phi_X^G:T^*X\rightarrow \mathfrak{g}^*\). As an application, the bigness of the tangent bundles of certain quasi-homogeneous varieties are verified, including symmetric varieties, horospherical varieties and equivariant compactification of commutative linear algebraic groups. Finally, we study in details the Fano manifolds \(X\)with Picard number \(1\) which is an equivariant compactification of a vector group \(\mathbb{G}_a^n\). In particular, we will determine the pseudoeffective cone of \(\mathbb{P}(T^*X)\) and show that the image of the projectivised moment map along the boundary divisor \(D\) of \(X\) is projectively equivalent to the dual variety of the VMRT of \(X\).
11. Fano manifolds with big tangent bundle: a characterisation of \(V_5\) (with Andreas Höring)
    [abstract]   [arXiv]   Collectanea Mathematica 74 (2023), 639–686.
    Let \(X\) be a Fano manifold with Picard number one such that the tangent bundle \(T_X\) is big. If \(X\) admits a rational curve with trivial normal bundle, we show that \(X\) is isomorphic to the del Pezzo threefold of degree five.
12. Fano foliations with small algebraic ranks
    [abstract]   [arXiv]   Advances in Mathematics 423 (2023), 109038, 32 pp.
    In this paper we study the algebraic ranks of foliations on \(\mathbb{Q}\)-factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the local positivity of the anti-canonical divisors of foliations, obtaining a lower bound for the algebraic rank of a foliation in terms of Seshadri constant. We describe those foliations whose algebraic rank slightly exceeds this bound and classify Fano foliations on smooth projective varieties attaining this bound. Finally we construct several examples to illustrate the general situation, which in particular allow us to answer a question asked by Araujo and Druel on the generalised indices of foliations.
13. Examples of Fano manifolds with non-pseudoeffective tangent bundle (with Andreas Höring and Feng Shao) - [Oberwolfach Report]
    [abstract]   [arXiv]   Journal of the London Mathematical Society 106 (2022), no. 1, 27-59.
    Let \(X\) be a Fano manifold. While the properties of the anticanonical divisor \(-K_X\) and its multiples have been studied by many authors, the positivity of the tangent bundle \(T_X\) is much more elusive. We give a complete characterisation of the pseudoeffectivity of \(T_X\) for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.
14. Stability of the tangent bundles of complete intersections and effective restriction
    [abstract]   [arXiv]   Annales de l'Institut Fourier 71 (2021), no. 4, 1601-1634.
    For \(n\geq 3\), let \(M\) be an \((n+r)\)-dimensional irreducible Hermitian symmetric space of compact type and let \(\mathcal{O}_M(1)\) be the ample generator of \(Pic(M)\). Let \(Y=H_1\cap\dots\cap H_r\) be a smooth complete intersection of dimension \(n\) where \(H_i\in\vert \mathcal{O}_M(d_i)\vert\) with \(d_i\geq 2\). We prove a vanishing theorem for twisted holomorphic forms on \(Y\). As an application, we show that the tangent bundle \(T_Y\) of \(Y\) is stable. Moreover, if $X$ is a smooth hypersurface of degree \(d\) in \(Y\) such that the restriction \(Pic(Y)\rightarrow Pic(X)\) is surjective, we establish some effective results for \(d\) to guarantee the stability of the restriction \(T_Y\vert_X\). In particular, if \(Y\) is a general hypersurface in \(\mathbb{P}^{n+1}\) and \(X\) is general smooth divisor in \(Y\), we show that \(T_Y\vert_X\) is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.
15. Strictly nef vector bundles and characterizations of \(\mathbb{P}^n\) (with Wenhao Ou and Xiaokui Yang)
    [abstract]   [arXiv]   Complex Manifolds 8 (2021), no. 1, 148-159.
    In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.
16. Fano manifolds containing a negative divisor isomorphic to a rational homogeneous space of Picard number one
    [abstract]   [arXiv]   International Journal of Mathematics 31 (2020), no. 9, 2050066, 14 pp.
    
    Let \(X\) be an \(n\)-dimensional complex Fano manifolds \((n\geq 3)\). Assume that \(X\) contains a divisor \(A\), which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle \(\mathscr{N}^*_{A/X}\) is ample over \(A\). Building on the works of Tsukioka, Watanabe and Casagrande-Druel, we give a complete classification of such pairs \((X,A)\).
    [erratum]    [PDF]    International Journal of Mathematics 33 (2022), no. 14, 2292003, 16 pp.
    
    In this paper, we make a correction to Theorem 1.2 of the aforementioned paper [J. Liu, Fano manifolds containing a negative divisor isomorphic to a rational homogeneousspace of Picard number one, Int. J. Math. 31(9) (2020) 2050066].
17. Note on quasi-polarized canonical Calabi-Yau threefolds
    [abstract]   [arXiv]   Comptes Rendus Mathématique. Académie des Sciences. Paris 358 (2020), no. 4, 415-420.
    Let \((X,L)\) be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that \(\vert mL\vert\) is basepoint free for \(m\geq 4\) . Moreover, if the morphism \(\Phi_{\vert 4L\vert}\) is not birational onto its image and \(h^0(X,L)\geq 2\), then \(L^3=1\). As an application, if \(Y\) is a \(n\)-dimensional Fano manifold such that \(-K_Y=(n-3)H\) for some ample divisor \(H\), then \(\vert mH\vert\) is basepoint free for \(m\geq 4\) and if the morphism \(\Phi_{\vert 4H\vert}\) is not birational onto its image, then \(Y\) is either a weighted hypersurface of degree \(10\) in the weighted projective space \(\mathbb{P}(1,\cdots,1,2,5)\) or \(h^0(Y,H)=n-2\).
18. Seshadri constants of the anticanonical divisors of Fano manifolds with large index
    [abstract]   [arXiv]   Journal of Pure and Applied Algebra 224 (2020), no. 12, 106438, 17 pp.
    Seshadri constants, introduced by Demailly, measure the local positivity of a nef divisor at a point. In this paper, we compute the Seshadri constants of the anticanonical divisors of Fano manifolds with coindex at most 3 at a very general point. As a consequence, if \(X\) is a nonsingular Fano threefold which is very general in its deformation family, then \(\varepsilon(X,-K_X;x)\leq 1\) for all points \(x\in X\)if and only if \(\vert-K_X\vert\) is not base point free.
19. Second Chern class of Fano manifolds and anti-canonical geometry
    [abstract]   [arXiv]   Mathematische Annalen 375 (2019), no. 1-2, 655–669.
    Let \(X\) be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of \(X\) in terms of its index and degree. As an application, if \(Y\) is a \(n\)-dimensional Fano manifold with \(-K_Y=(n-3)H\) for some ample divisor \(H\), we prove that \(h^0(Y,H)\geq n-2\). Moreover, we show that the rational map defined by \(\mid mH\mid\) is birational for \(m\geq 5\), and the linear system \(\mid mH\mid\) is basepoint free for \(m\geq 7\). As a by-product, the pluri-anti-canonical systems of singular weak Fano varieties of dimension at most \(4\) are also investigated.
20. Characterization of projective spaces and \(\mathbb P^r\)-bundles as ample divisors
    [abstract]   [arXiv]   Nagoya Mathematical Journal 233 (2019), 155–169.
    Let \(X\) be a projective manifold of dimension \(n\). Suppose that \(T_X\) contains an ample subsheaf. We show that \(X\) is isomorphic to \(\mathbb{P}^n\). As an application, we derive the classification of projective manifolds containing a \(\mathbb{P}^r\)-bundle as an ample divisor by the recent work of Litt.

Others

21. Finite covering of intersections of two quadrics
    Appendix to "Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number \(1\)" by Feng Shao and Guolei Zhong
    [abstract]   [arXiv]   Preprint (2023)
    We show that the tangent bundle of a general finite cover of a general complete intersection of two quadrics is not pseudo-effective.
22. Projective manifolds containing an ample divisor isomorphic to a locally rigid Fano complete intersection (not intended for publication)
    [abstract]    [PDF]    Preprint (2019)
    In this short note, we classify the following pair \((X,A)\): \(X\) is a projective manifold and \(A\) is an ample divisor on \(X\) such that \(A\) is isomorphic to a smooth locally rigid Fano complete intersection in a rational homogeneous space S of Picard number one.