An application of the canonical bundle formula

Frobenius pull backs of principal $G$-bundles and their canonical parabolics

The paper is withdrawn.

On foliations with nef anti-canonical bundle

In this paper we prove that the anti-canonical bundle of a holomorphic foliation $\mathcal{F}$ on a complex projective manifold cannot be nef and big if either $\mathcal{F}$ is regular, or $\mathcal{F}$ has a compact leaf. Then we address codimension one regular foliations whose anti-canonical bundle is nef with maximal Kodaira dimension.

Divisor classes and the virtual canonical bundle for genus 0 maps

Some divisor class relations are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target.

Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

We investigate the deformations of pairs $(X,L)$, where $L$ is a line bundle on a smooth projective variety $X$, defined over an algebraically closed field $\mathbb{K}$ of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair $(X,L)$ is homotopy abelian whenever $X$ has trivial canonical bundle, and so these deformations are unobstructed.

Rank two vector bundles with canonical determinant

Denote by B^k_{2,K} the locus of vector bundles of rank two and canonical determinant. We show that for a generic curve of genus g, B^k_{2,K} is non-empty if g is sufficiently large.

The moduli space of stable supercurves and its canonical line bundle

We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.

A canonical bundle formular of projective Lagrangian fibrations

We classify singular fibres of a projective Lagrangian fibration over codimension one points. As an application, we obtain a canonical bundle formula for a projective Lagrangian fibration over a smooth manifold.

On the first Gaussian map for Prym-canonical line bundles

We prove by degeneration to Prym-canonical binary curves that the first Gaussian map of the Prym canonical line bundle $ω_C \otimes A$ is surjective for the general point [C,A] of R_g if g >11, while it is injective if g < 12.

On a generalized canonical bundle formula for generically finite morphisms

We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with $\mathbb{R}$-coefficients). This complements Filipazzi's canonical bundle formula for morphisms with connected fibres. It is then applied to obtain a subadjunction formula for log canonical centers of generalized pairs. As another application, we show that the image of an anti-nef log canonical generalized pair has the structure of a numerically trivial log canonical generalized pair. This readily implies a result of Chen--Zhang. Along the way we prove that the Shokurov type convex se…

A travel guide to the canonical bundle formula

We survey known results on the canonical bundle formula and its applications in algebraic geometry.

Petri map for rank two bundles with canonical determinant

We prove the Bertram-Feinberg-Mukai conjecture for a generic curve $C$ of genus $g$ and a semistable vector bundle $E$ of rank two and determinant $K$ on $C$, namely we prove the injectivity of the Petri-canonical map $S^2(H^0(E))\to H^0(S^2(E))$.

Manifolds of low dimension with trivial canonical bundle in Grassmannians

We study fourfolds with trivial canonical bundle which are zero loci of sections of homogeneous, completely reducible bundles over ordinary and classical complex Grassmannians. We prove that the only HyperK{ä}hler fourfolds among them are the example of Beauville and Donagi, and the example of Debarre and Voisin. In doing so, we give a complete classification of those varieties. We include also the analogous classification for surfaces and threefolds.

Prym map and second gaussian map for Prym-canonical line bundles

We show that the second fundamental form of the Prym map lifts the second gaussian map of the Prym-canonical bundle. We prove, by degeneration to binary curves, that this gaussian map is surjective for the general point [C,A] of R_g for g > 19.

Picard数为1的Fano簇上切丛典范扩张的稳定性

Stability of the canonical extension of tangent bundles on Picard-rank-1 Fano varieties

本文研究Picard数为1的Fano簇上切丛典范扩张的斜率稳定性，证明在指数整除维数或维数加一时切丛稳定性蕴含典范扩张的（半）稳定性，并给出其在曲线稳定向量丛模空间上的应用推论。

A Petri Theorem for Rank 2 Bundles with Canonical Determinant

This paper establishes the correctness of a conjecture of Bertram-Feinberg and Mukai for a special class of globally generated rank-two bundles with canonical determinant over a generic Riemann surface of genus at least four.

The normal bundle of canonical genus 8 curves

We study the stability of the normal bundle of canonical genus $8$ curves and prove that on a general curve the bundle is stable. The proof rests on Mukai's description of these curves as linear sections of a Grassmannian $\mathrm{G}(2,6)$. This is the next case of a conjecture by M. Aprodu, G. Farkas, and A. Ortega: the general canonical curve of every genus $g \geq 7$ should have stable normal bundle. We also give some more evidence for this conjecture in higher genus.

Invariance for multiples of the twisted canonical bundle

In this paper, we show an extension type theorem for twisted pluricanonical sections on a family of smooth projective manifolds (the twisting line bundle being pseudo-effective and having a prescribed multiplier ideal on the central fiber).

On The Projective Normality And Normal Presentation On Higher Dimensional Varieties With Nef Canonical Bundle

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.

$\mathbb{P}^3$ 的典范线丛上的稳定性条件

Stability conditions on the canonical line bundle of $\mathbb{P}^3$

本文研究三维射影空间上典范线丛全空间的稳定性条件空间，构造了几何稳定性条件族及其代数边界子集，并利用球面扭转构造其他稳定性条件。

A remark on the decomposition theorem for direct images of canonical sheaves tensorized with semipositive vector bundles

The purpose of this short note is to give a remark on the decomposition theorem for direct images of canonical sheaves tensorized with Nakano semipositive vector bundles. Although our result is a direct consequence of Takegoshi's work, it was not stated explicitly in his article. Here we give the precise statement and the proof.

有正伴随典范线丛的形变

Deformation of nef adjoint canonical line bundles

本文研究射影与Kähler族中典范线丛的nef性、半充足性形变性质，证明伴随丛nef性与半充足性的形变不变性，推广广义多重亏格不变性，并部分证实三维Kähler形变下典范线丛半充足性的稳定性猜想。

Nef and abundant divisors, semiampleness and canonical bundle formula

In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$ effective, $L \in Pic (X)$ nef, nefness and abundance of $K_X+B+L$ implies semiampleness. This essentially generalizes Kawamata's theorem to the setting of generalized abundance.

附录：关于广义典范丛公式和复解析环境下补集有界性

Addendum: On generalized canonical bundle formula and boundedness of complements in complex analytic setting

本文在复解析环境下，无需nef部分假设，建立了广义lc平凡纤维化的广义典范丛公式，并给出了相应的代数版本。

关于反典范丛的两个非消失性结果

Two Non-Vanishing results concerning the Anti-Canonical Bundle

本文研究了klt三维fold对在反log典范丛为nef时的非消失性问题，证明了其Kodaira维数非负，并通过建立任意维度均成立的等变非消失性结果来实现这一目标。

Bounds on the denominators in the canonical bundle formula

In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose generic fibre is a rational curve. In particular we find a bound for the denominators of the discriminant and the moduli divisor.

On a generalized canonical bundle formula and generalized adjunction

In this note, we extend the theories of the canonical bundle formula and adjunction to the case of generalized pairs. As an application, we study a particular case of a conjecture by Prokhorov and Shokurov.

Fibrations on four-folds with trivial canonical bundles

Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and prove that there is no such fibration in the sixth class. We classify all such fibrations whose generic fibre is the Jacobian of a genus two curve.

Base point free theorems--saturation, b-divisors, and canonical bundle formula--

We reformulate base point free theorems. Our formulation is flexible and has some important applications. One of the main purposes of this paper is to prove a generalization of the base point free theorem in Fukuda's paper: On numerically effective log canonical divisors.

亏格至少为7的一般典范曲线的法丛是半稳定的

The normal bundle of a general canonical curve of genus at least 7 is semistable

本文研究任意特征代数闭域上一般典范曲线的法丛稳定性，证明亏格不为4、6时法丛半稳定，且亏格模6余1或3时法丛稳定，将亏格下界推进至7。

一般射影空间上线丛的规范处理

A canonical treatment of line bundles over general projective spaces

本文对一般域上有限维向量空间的射影空间及其线丛理论进行了无基的规范处理，并特别考察了全局正则截面空间，同时指出域的代数闭性起作用之处。

On negativity of total $k$-jet curvature and ampleness of the canonical bundle

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 $X$.

余切丛在具有丰富典范类的曲面上的稳定性

Stability of the Cotangent Bundle on Surfaces with Ample Canonical Class

本文研究K3曲面及具有丰富典范类的曲面上余切丛的Bridgeland稳定性，找到了阈值$t_0$使得余切丛在$t>t_0$时半稳定，并探讨了伪射影平面和伪二次曲面等特例。

Ordinary varieties with trivial canonical bundle are not uniruled

We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer's results, implies that varieties of the above type have strongly semistable tangent bundles with respect to any polarization.

叶状结构、斜率稳定性及Deligne-Mumford栈上对数典范丛的正性

Foliations, slope stability, and positivity of log canonical bundles on Deligne-Mumford stacks

将Campana-Păun关于光滑射影簇上叶状结构、斜率稳定性及对数典范丛正性的结果推广到具有射影粗模空间的光滑真DM栈上，作为将Viehweg双曲性结果推广到DM栈及特定KSBA模空间系列论文的第三部分。

Shokurov's conjecture on conic bundles with canonical singularities

A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $X\to Z$ is a conic bundle such that $X$ has canonical singularities and $Z$ is $\mathbb{Q}$-Gorenstein, then $Z$ is always $\frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $\frac{1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative …

The Period-Index Problem of the Canonical Gerbe of Symplectic and Orthogonal Bundles

We consider regularly stable parabolic symplectic and orthogonal bundles over an irreducible smooth projective curve over an algebraically closed field of characteristic zero. The morphism from the moduli stack of such bundles to its coarse moduli space is a $μ_2$-gerbe. We study the period and index of this gerbe, and solve the corresponding period-index problem.

A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles

In this paper I present a new geometric approach to the factorization rule for generalised theta functions. Let $X$ be an irreducible projective nodal curve with one singularity and let $Y$ be its normalization. Recently I have constructed the moduli stack $GVB(X)$ of rank $n$ Gieseker vector bundles on $X$ and have shown that its normalization is a locally trivial fibration over the moduli stack $VB(Y)$ of vector bundles on $Y$, where the fibre is a canonical compactification of $Gl_n$. In this paper I prove a canonical direct sum decomposition of the space of global sections of a power of th…

$n$-excisive functors, canonical connections, and line bundles on the Ran space

Let $X$ be a smooth algebraic variety over $k$. We prove that any flat quasicoherent sheaf on $\operatorname{Ran}(X)$ canonically acquires a D-module structure. In addition, we prove that, if the geometric fiber $X_{\overline{k}}$ is connected and admits a smooth compactification, then any line bundle on $S \times \operatorname{Ran}(X)$ is pulled back from $S$, for any locally Noetherian $k$-scheme $S$. Both theorems rely on a family of results which state that the (partial) limit of an $n$-excisive functor defined on the category of pointed finite sets is trivial.

关于具有平凡典范丛的拟\(F -\)分裂簇的整体和局部几何

On the global and local geometry of quasi-$F$-split varieties with trivial canonical bundle

研究具有平凡典范丛的拟\(F -\)分裂簇，通过多种方法证明正则拟\(F^{\infty}-\)分裂簇性质、解答相关曲面问题、证明正规簇几何正规性，以及在奇点解消假设下证明特定拟\(F^e -\)纯正规簇是对数典范的。

典范丛公式与Schnell关于某类代数纤维空间的猜想

Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell

本文利用典范丛公式，在两种独立假设下证明了Schnell关于代数纤维空间的猜想，其中一种假设涉及刚性流表示的典范类。

关于广义canonical包公式和复分析设定中补集的有界性

On generalized canonical bundle formula and boundedness of complements in complex analytic setting

本文研究了复分析设定中广义lc-平凡纤维化中的广义canonical包公式，探讨了判别b-除子和模b-除子的兼容性及补集的有界性。

关于特征 $p>0$ 下 Fano 簇的典范丛公式与有效双有理性

On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。

曲线半稳定 Higgs 丛与典范丛丰富的光滑射影簇

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。

Orbifold generic semi-positivity: an application to families of canonically polarized manifolds

In this article we establish a version of Y. Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs. As an application, we show that the canonical bundle associated to a lc pair is big as soon as there exists a generically injective morphism from an ample line bundle to some symmetric power of the cotangent bundle associated to the orbifold pair.

典范扩展的 nef 切包 manifold

Canonical extensions of manifolds with nef tangent bundle

本文证明了当流形的切包为 nef 时，其典范扩展是 Stein 流形，并探讨了相关等价性问题。

Restricted Lazarsfeld-Mukai bundles and canonical curves

We prove two results. First, we establish that the normal bundle of any smooth curve of genus 7 having maximal Clifford index is stable. Note that 7 is the smallest genus for which such a result could possibly hold. We then show that rank four Lazarsfeld-Mukai vector bundles on a curve that lies on a general K3 surface are stable. Both results have consequences for Mercat's conjecture on higher rank vector bundles on generic curves.

Canonical singular hermitian metrics on relative canonical bundles

We introduce a new class of canonical AZD's (called the supercanonical AZD's) on the canonical bundles of smooth projective varieties with pseudoeffective canonical classes. We study the variation of the supercanonical AZD $\hat{h}_{can}$ under projective deformations and give a new proof of the invariance of plurigenera.

Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber $F$ has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when $F$ is $F$-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases.

Nonexistence of semiorthogonal decompositions and sections of the canonical bundle

For any admissible subcategory of the bounded derived category of coherent sheaves on a smooth proper variety, we prove that sections of the canonical bundle impose a strong constraint on the supports of the objects of the subcategory or its semiorthogonal complement. We also show that admissible subcategories are rigid under the actions of topologically trivial autoequivalences. As applications of these results, we prove that the derived category of various minimal varieties in the sense of the minimal model program admit no non-trivial semiorthogonal decompositions, generalizing the result f…

Canonical singular hermitian metrics on relative logcanonical bundles

This supersedes 0704.0566. We prove the invariance of logarithmic plurigenera for a projective family of KLT pairs and the adjoint line bundle of KLT line bundles. The proof uses the canonical singular hermitian metrics on relative logcanonical bundles.

Positivity of relative canonical bundles for families of canonically polarized manifolds

Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Main application is an analytic proof of the quasi-projectivity of the moduli space of canonically polarized manifolds. Further applications in arXiv:1002.4858v2.

Negative Holomorphic curvature and positive canonical bundle

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Ampleness of the CM line bundle on the moduli space of canonically polarized varieties

We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.

Compact Kähler threefolds with non-nef canonical bundle and symplectic geometry

We show that the nefness of the canonical bundle of compact Kähler threefolds is invariant under deformed symplectic diffeomorphisms.