An algebraic proof of K-semistability of the projective plane

Semistability of Lazarsfeld-Mukai bundles via parabolic structures

Our aim in this article is to produce new examples of semistable Lazarsfeld- Mukai bundles on smooth projective surfaces $X$ using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld-Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of $X$ corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld-Mukai bundles. This gives semistable Lazarsfeld-Mukai bundles on Kawamata covers of the projective plane and …

Semistability and restrictions of tangent bundle to curves

We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,φ)$, where $C$ is a compact connected Riemann surface and $φ: C\to X$ a holomorphic map, such that the pull back $φ^*TX$ is not semistable. 2. The variety $X$ admits an étale covering by an abelian variety. 3. The dimension $\dim X \leq 1$. We conjecture that all complex projective manifolds are of the above type, and prove that the following classes are among those that are of the above type. i) All $X$ with a finite fundamental group. ii) All $X$ such that t…

Semistability vs. nefness for (Higgs) vector bundles

According to Miyaoka, a vector bundle E on a smooth projective curve is semistable if and only if a certain numerical class in the projectivized bundle PE is nef. We establish a similar criterion for the semistability of Higgs bundles: namely, such a bundle is semistable if and only if for every integer s between 0 and the rank of E, a suitable numerical class in the scheme parametrizing the rank s locally-free Higgs quotients of E is nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the above mentioned classes is equivalent to th…

Postnikov-Stability versus Semistability of Sheaves

We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of semistable sheaves. As one application we compactify a moduli space of stable bundles using genuine complexes via Fourier-Mukai transforms.

On two Notions of Semistability

We show that certain semistable sheaves on the projective plane with linear Hilbert polynomial are cokernels of semistable morphisms of decomposable sheaves.We exhibit certain locally closed subvarieties of moduli spaces of semistable sheaves as quotients modulo nonreductive groups.

Semistability of restricted tangent bundles and a question of I. Biswas

Let M be a complex projective manifold with the property that for any compact Riemann surface C and holomorphic map f: C -> M the pullback of the tangent bundle of M is semistable. We prove that in this case M is a curve or a finite etale quotient of an abelian variety answering a conjecture of I. Biswas.

Reduction of the Semistability Condition for Tensors

In this article we study a special class of vector bundles, called tensors. A tensor consists of a vector bundle $E$ over a smooth irreducible projective variety and a morphism of vector bundles $φ$. As for classical vector bundles, there exists a notion of stability for these objects given in terms of filtrations of the vector bundle $E$. The aim of the present paper is to prove that if a destabilizing filtration is "too" long then there exists a shorter subfiltration which destabilizes as well. Moreover, we describe some related combinatorial problems, which arise from the description of a t…

Semistability of Frobenius direct images over curves

Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Given a semistable vector bundle $E$ over $X$, we show that its direct image $F\_*E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank-$p$ vector bundles $F\_*L$, where $L$ is a line bundle over $X$.

沿线性子空间爆破的射影空间的 Seshadri 斜率 $\K$-半稳定性

Seshadri slope $\K$-semistability for the blow-up of projective space along a linear subspace

本文应用 Chow-Mumford 线丛理论，证明了沿线性子空间爆破的射影空间在例外除子上的形变到法锥测试构型具有 $\K$-半稳定性。

An obstruction to asymptotic semistability and approximate critical metrics

In this paper, we consider an obstruction to asymptotic Chow-semistability of a polarized Kaehler algebraic manifold. Even when a linear algebraic group of positive dimension acts nontrivially and holomorphically on a polarized Kaehler algebraic manifold with constant scalar curvature, the vanishing of the obstruction allows us to generalize Donaldson's construction of approximate solutions for equations of balanced metrics.

关于奇异射影超曲面半稳定性的一个注记

A note on the semistability of singular projective hypersurfaces

本文利用次数、奇点最大重数与奇异轨迹维数，并结合 Benoist 的交维数下界及 Hilbert–Mumford 判据，给出奇异射影超曲面半稳定的充分条件。

关于射影丛爆破的K-稳定性

On the K-stability of blow-ups of projective bundles

研究法诺簇上射影线丛爆破的K-稳定性，证明当除子B与2L成比例时，爆破的K-半稳定和K-多稳定等价于对数法诺对的相应性质，并给出系数；当比例非2时爆破K-不稳定。

Positivity vs. slope semistability for bundles with vanishing discriminant

We give an algebraic proof valid in arbitrary characteristic for the known equivalence between (strongly) slope semistable vector bundles with vanishing discriminant and vanishing determinant and numerically flat bundles. We also address a related question of Misra.

射影环面簇的数值半稳定性

Numerical semistability of projective toric varieties

本文给出光滑环面簇数值半稳定的充要条件，并证明极化环面簇的渐近数值半稳定性等价于对环面退化的K-半稳定性。

Semistability of invariant bundles over $G/Γ$

Let $G$ be a connected reductive affine algebraic group defined over $\mathbb C$, and let $Γ$ be a cocompact lattice in $G$. We prove that any invariant bundle on $G/Γ$ is semistable.

Symplectic stability, analytic stability in non-algebraic complex geometry

We give a systematic presentation of the stability theory in the non-algebraic Kaehlerian geometry. We introduce the concept of "energy complete Hamiltonian action". To an energy complete Hamiltonian action of a reductive group G on a complex manifold one can associate a G-equivariant maximal weight function and prove a Hilbert criterion for semistability. In other words, for such actions, the symplectic semistability and analytic semistability conditions are equivalent.

On the behaviour of strong semistability in geometric deformations

Let $Y \to B$ be a relative smooth projective curve over an affine integral base scheme $B$ of positive characteristic. We provide for all prime characteristics example classes of vector bundles $\mathcal{S}$ over $Y$ such that $\mathcal{S}$ is generically strongly semistable and semistable but not strongly semistable for some special fibre. This also provides new examples of the behaviour of Hilbert-Kunz multiplicities in geometric families.

Smoothness, Semistability, and Toroidal Geometry

We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings. Needless to say, this theorem is a weak version of Hironaka's well known theorem on resolution of singularities. Our proof has the feature that it builds on two standard techniques of algebraic geometry: semistable reduction for curves, and t…

Projectivity of moment map quotients

Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding moment map. It is well known that the quotient M/K is a complex space in a natural way. We prove that M/K is a projective variety. In particular, it follows that semistability with respect to a moment map is equivalent to semistability in the sense of Mumford.

Optimal destabilizing centers and equivariant K-stability

We give an algebraic proof of the equivalence of equivariant K-semistability (resp. equivariant K-polystability) with geometric K-semistability (resp. geometric K-polystability). Along the way we also prove the existence and uniqueness of minimal optimal destabilizing centers on K-unstable log Fano pairs.

Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is f…

Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Let $E$ be a vector bundle over a smooth curve $C$, and $S = \mathbb{P} E$ the associated projective bundle. We describe the inflectional loci of certain projective models $ψ\colon S \dashrightarrow \mathbb{P}^n$ in terms of Quot schemes of $E$. This gives a geometric characterisation of the Segre invariant $s_1 (E)$, which leads to new geometric criteria for semistability and cohomological stability of bundles over $C$. We also use these ideas to show that for general enough $S$ and $ψ$, the inflectional loci are all of the expected dimension. An auxiliary result, valid for a general subvarie…

射影超曲面束、Griffiths高度与几何不变量理论 I

Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. I

本文通过GIT高度界定射影超曲面束的Griffiths高度，验证Kato-Koshikawa猜想的几何版本，并推广至半齐次奇点情形，证明稳定Griffiths高度的下半连续性。

Moduli Spaces of Coherent Sheaves on Projective Deligne-Mumford Stacks over Algebraic Spaces

In this paper, we study the geometric invariant theory on algebraic spaces, and construct te moduli spaces of $\mathcal{H}$-semistable sheaves on projective Deligne-Mumford stacks over algebraic spaces $S$. We prove that this moduli space is projective over $S$ as an algebraic space.

On the density of foliations without algebraic solutions on weighted projective planes

We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.

Picard秩为2和3的光滑射影环面三维体中曲面的代数双曲性

Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

本文研究Picard秩为2和3的光滑射影环面三维体中曲面的代数双曲性，利用组合描述与Haase和Ilten的方法，在这些簇中确定了众多代数双曲性曲面。

Algebraic varieties with quasi-projective universal cover

We prove that the universal cover of a normal, projective variety X is quasi-projective if and only if a finite, étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.

加权射影空间中非常一般超曲面的代数双曲性

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。

最小的代数空间填充曲线在射影线的积上

Minimal algebraic space filling curves on the product of projective lines

本文研究了射影线积上最小次数的光滑代数空间填充曲线，证明了存在大量例子并扩展了Homma和Kim的结果。

A birational embedding of an algebraic curve into a projective plane with two Galois points

A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.

How to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces

A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree $d$ in $CP^n$ is described. Using this method, rational cohomology groups of such spaces with $n=2, d \le 4$ and $n=3=d$ are calculated.

On semistable principal bundles over a complex projective manifold

Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by χis numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he…

Classification of two-dimensional algebraic projective semigroups

In this article, we address the classification of smooth projective algebraic surfaces over complex numbers admitting algebraic semigroup structures. We give a full description of those surfaces $S$, which has at least one non-trivial algebraic semigroup structure, when the Kodaira dimension of $S$ is $ -\infty$ and $ 0$. For the case "$ κ(S)=1$", we give a description of one special type of elliptic surfaces which admit non-trivial algebraic semigroup laws. \\ For a given surface $S$, it is an interesting problem to describe all algebraic semigroup structures on it and determine the dimension…

关于射影代数簇的内射映射

On Endomorphisms of Projective Algebraic Varieties

本文研究了射影簇内射映射的代数动力学，通过刻画迭代像、Stein分解和稳定性现象，完成Brion的结果。

射影空间中非常一般超曲面的代数双曲性

Algebraic hyperbolicity of very general hypersurfaces in projective spaces

本文证明了射影空间中非常一般超曲面代数双曲性的最后未解情形，即四次三维簇，完成了分类，方法基于Coskun--Riedl构造及稳定映射模空间和对数几何。

A family of flat connections on the projective space having dihedral monodromy and algebraic Garnier solutions

A. Girand has constructed an explicit two-parameter family of flat connections over the complex projective plane $\mathbb{P}^2$. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a conic and three tangent lines. In this paper, we give a generalization of this construction. That is, we construct an explicit $n$-parameter family of flat connections over the complex projective space $\mathbb{P}^n$. Moreover, we discuss the relation between these connections and the Garnier system.

Algebraic varieties with semialgebraic universal cover

We study projective varieties whose universal cover is biholomorphic to a semialgebraic open subset of a projective variety.

Torsion cohomology classes and algebraic cycles on complex projective manifolds

Atiyah and Hirzebruch gave examples ofeven degree torsion classes in the singularcohomology of a smooth complex projective manifold, which arenot Poincaré dual to an algebraiccycle. We notice that the order ofthese classes are small comparedto the dimension of the manifold.However, building upon a construction ofKollàr, one can provide such examples witharbitrary high prime order, the dimension being fixed. This method alsoprovides examples of torsion algebraiccycles, which are non trivial in the Griffiths' groups, and lie in a arbitrary high level of the H.Saito filtration onChow groups.

A Compactification of the Space of Algebraic Maps from P^1 to P^n

We provide a natural smooth projective compactification of the space of algebraic maps from the projective line to the projective space of dimension n by adding a divisor with simple normal crossings.

Algebraic entropy for smooth projective varieties

We show that the spectral radius for the action of a self map $f$ of a smooth projective variety (over an arbitrary base field) on its $\ell$-adic cohomology is achieved on the $f^*$-stable sub-algebra generated by any ample class. This generalizes a result of Esnault-Srinivas who had obtained an analogous result for automorphisms of surfaces. Over $\mathbb{C}$ we also show that this sub-algebra is naturally an irreducible representation of a Looijenga-Lunts-Verbitsky type Lie algebra acting on the cohomology of a smooth projective variety.

实代数簇的相对形式及具有实形式代数模的拟射影曲面的例子

Relative forms of real algebraic varieties and examples of quasi-projective surfaces with algebraic moduli of real forms

本文提出一个框架以精确化“由簇参数化的簇的实形式族”概念，并构造了首个具有任意大代数模维度的非同构实形式的拟射影实曲面例子。

Homological Projective Duality

We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties $X$ and $Y$ in dual projective spaces are Homologically Projectively Dual, then we prove that the orthogonal linear sections of $X$ and $Y$ admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate Homological Projective Duality for projectivizations of vec…

光滑射影超曲面的上同调上复共轭的代数描述

Algebraic description of complex conjugation on cohomology of a smooth projective hypersurface

本文利用 Griffiths 的 Jacobian 环描述，刻画数域上光滑射影超曲面本原中间维数 de Rham 上同调上的复共轭，并指出其除显式周期因子外具有代数性。

Rationality of cycles on function field of exceptional projective homogeneous varieties

In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type $F_4$ or $E_8$ and over the base field, which can be of any characteristic.

On the geometry of the Hilbert schemes of points in the projective plane

We classify minimal-degree curves in the Hilbert schemes of points on algebraic surfaces. When the algebraic surface is the projective plane, the nef cone and a flip structure of these Hilbert schemes are determined.

Projective Normality Of Algebraic Curves And Its Application To Surfaces

Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<°L\le 2g-5$. Then $L$ is normally generated if $°L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C'$ with $C\stackrelϕ\to C'$ and $D$ a divisor on $C'$ with $4p<°D< \frac{g-1}{6}-2p$. Then $K_C(-ϕ^*D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.

Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

We study the algebraic hyperbolicity of very general hypersurfaces in $\mathbb{P}^m \times \mathbb{P}^n$ by using three techniques that build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As a result, we completely answer the question of whether or not a very general hypersurface of bidegree $(a,b)$ in $\mathbb{P}^m \times \mathbb{P}^n$ is algebraically hyperbolic, except in $\mathbb{P}^3 \times \mathbb{P}^1$ for the bidegrees $(a,b)= (7,3), (6,3)$ and $(5,b)$ with $b\geq 3.$ As another application of these techniques, we improve the known result that very general hypers…

双标作用于射影平面的代数磁不变量

Algebraic magnetism invariants of a double scalar action on the projective plane

本文研究双标作用于射影平面的代数磁不变量，计算纯磁体和吸引子，并引入lambda可实现磁体的概念，同时解决了一些猜想。

On automorphisms and endomorphisms of projective varieties

We first show that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal projective variety. Then we show that very few connected algebraic semigroups can be realized as endomorphisms of some projective variety X, by describing the structure of all connected subsemigroup schemes of End(X).

用代数射影几何求解吉布斯问题

Solving the Gibbs Problem with Algebraic Projective Geometry

本文提出用代数射影几何求解吉布斯问题，即从三个位置矢量进行轨道确定，给出一种新颖的以原点为焦点的椭圆代数拟合方法，结果与经典矢量分析法一致。

Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces

The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f:{\mathbb C}\to X$. Using the formalism of directed varieties, we prove here that this assertion holds true in case $X$ satisfies a strong general type condition that is related to a certain jet-semistability property of the tangent bundle $T\_X$. We then use this fact to confirm a long-standing conjecture of Kobayashi (1970), according to which a very general algebraic hypersurface…

具有大反典范类的簇的K-稳定性

K-stability for varieties with a big anticanonical class

将代数K-稳定性理论推广到具有大反典范类的射影klt对，并证明K-半稳定性条件迫使它们具有klt反典范模型，且稳定性性质与原对相同。

Computing projective equivalences of special algebraic varieties

This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets on the projective line. In particular, we design a unifying approach that finds for two algebraic varieties $X,Y$ from special classes an associated set of automorphisms of the projective line (the so called good candidate set) consisting of candidates for the construction of possible mappings $X\rightarrow Y$. The functionality of the designed method is pres…

Spaces of algebraic maps from real projective spaces into complex projective spaces

We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural "degree" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions.