Kang Zuo
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
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A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle $K_X$ of a Kobayashi hyperbolic compact complex manifold $X$ is ample. In this note we prove that $K_X$ is ample if $X$ is projective and satisfies a stronger condition of nondegenerate negative total $k$-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on $X$.
In this paper, we study a variation of a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.
Positivity, plethysm and hyperbolicity of Siegel varieties in positive characteristic
本文研究了特征 $p$ 下Siegel modular variety的双曲性，利用Schur函子的plethysm运算和新的 $(φ,D)$-ample正性概念，证明了多种自守向量丛满足该正性，推广了Hodge线丛的已知结论。
Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties…
Algebraic hyperbolicity of very general hypersurfaces in weighted projective spaces
本文研究加权射影空间中非常一般超曲面的代数双曲性，通过给出丰沛线丛倍数 $m$ 的界，证明了相应零点轨迹具有代数双曲性。
In this paper, we introduce a sub-family of the usual generalized Wronskians, that we call geometric generalized Wronskians. It is well-known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global f…
In this paper we examine different problems regarding complete intersection varieties of high degree in a complex projective space. First we show how one can deduce hyperbolicity for generic complete intersection of high multidegree and high codimension from the known results on hypersurfaces. Then we prove a existence theorem for jet differentials that generalizes a theorem of S. Diverio. Finally, motivated by a conjecture of O. Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove th…
For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-Păun. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbol…
Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion f…
This survey article mainly addresses to graduate students and young researchers in complex geometry willing to enter the beautiful word of connections between curvature and Kobayashi hyperbolicity. It is a detailed account of a recent breakthrough by Wu and Yau, shortly after generalized by Tosatti and Yang (and others), which sits on the crossroad between complex differential geometry and Kobayashi hyperbolicity. More specifically, an old conjecture by Kobayashi, stated at the very beginning of the theory, predicts that a compact hyperbolic manifold should have ample canonical bundle. Now, on…
Hyperbolicity of smooth logarithmic and orbifold pairs in $\mathbb{P}^n$
研究$\mathbb{P}^n$中超平面排列的条件，以确定关联的对数余切丛在边界上是 ample 的，并扩展到轨道设置，应用超曲面性结果，改进Darondeau-Rousseau的结论。
Algebraic hyperbolicity of adjoint linear systems on spherical varieties
论文证明了在球面变种上，伴随线系统非常一般的元素具有代数双曲性，并推导出该猜想在horospherical变种和toroidal球面变种上成立。
Curve semistable Higgs bundles and smooth projective varieties whose canonical bundle is ample
本文研究典范丛丰富的光滑射影簇上的 Simpson 系统，证明其 Higgs 丛稳定性及 Guggenheimer–Yau 不等式，并在等号时得到判别类消失与曲线半稳定性，同时给出代数双曲性及正实常数下界。
A hyperbolicity conjecture for adjoint bundles
本文提出了关于伴随丛的双曲性猜想，并在光滑射影环面簇情形下证实了该猜想，同时证明了 Gorenstein 环面簇的伪双曲性及三维簇的双曲性。
Hyperbolicity of adjoint linear series on varieties with positive tangent bundle
本文研究了在具有正切包的变种上，余线性系列的超曲性，证明了对于不同类别的极化对，当m足够大时，线性系统具有超曲性。
Varieties with ample Frobenius-trace kernel
研究正则变种上福布斯迹核的正性，发现其在射影空间中为 ample，但仅在 Picard 等级为 1 的 Fano 变种上成立。
Given a log canonical pair $(X, Δ)$, we show that $K_X+Δ$ is nef assuming there is no non-constant map from the affine line with values in the open strata of the stratification induced by the non-klt locus of $(X, Δ)$. This implies a generalization of the Cone Theorem where each $K_X+Δ$-negative extremal ray is spanned by a rational curve that is the closure of a copy of the affine line contained in one of the open strata of $\mathrm{Nklt}(X, Δ)$. Moreover, we give a criterion of Nakai type to determine when under the above condition $K_X+Δ$ is ample and we prove some partial results in the ca…
Consider a smooth projective family of complex polarized manifolds with semi-ample canonical sheaf over a quasi-projective manifold $V$. When the associated moduli map $V\to P_h$ from the base to coarse moduli space is quasi-finite, we prove that the generalized big Picard theorem holds for the base manifold $V$: for any projective compactification $Y$ of $V$, any holomorphic map $f:Δ-\{0\}\to V$ from the punctured unit disk to $V$ extends to a holomorphic map of the unit disk $Δ$ into $Y$. This result generalizes our previous work on the Brody hyperbolicity of $V$ (i.e. there are no entire cu…
As a result of our study of the hyperbolicity of the moduli space of polarized manifold, we give a general big Picard theorem for a holomorphic curve on a log-smooth pair $(X,D)$ such that $W=X\setminus D$ admits a Finsler pseudometric that is strongly negatively curved when pulled back to the curve. We show, by some refinements of the classical Viehweg-Zuo construction, that this latter condition holds for the base space $W$, if nonsingular, of any algebraic family of polarized complex projective manifolds with semi-ample canonical bundles whose induced moduli map $ϕ$ to the moduli space of s…
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a non-constant map to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration facto…
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