Georg Hein
We define generalizations of the Albanese variety for a projective variety X. The generalized Albanese morphisms X --> Alb_r(X) contract those curves C in X for which the induced morphism Hom(π_1(X),U(r)) --> Hom(π_1(C),U(r)) has a finite image. Thus, they may be interpreted as a U(r)-version of the Shafarevich morphism.
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In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is $F$-pure. Furthermore, we consider a notion of $F$-splitting for morphisms, and investigate it of the Albanese morphisms. We show that an $F$-split variety has $F$-split Albanese morphism, and that the $F$-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.
On quasi-Albanese morphisms for log canonical Calabi-Yau pairs
本文研究了log canonical Calabi-Yau对的准阿贝尔 morphisms，提出了toric pairs的特征化结果。
The Albanese morphism for hyperelliptic varieties
本文用阿贝尔簇A和有限群G具体刻画了超椭圆簇X的Albanese态射及其纤维结构，证明纤维为阿贝尔簇或超椭圆簇，并由此在某些情形下证明X的导出范畴不可分解。
For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they are connected and contractible.
Varieties with nef anticanonical divisors and Albanese morphisms of relative dimension one in positive characteristic
本文研究正特征代数闭域上具有 nef 反典范除数的光滑射影簇，并在其到 Albanese 簇的态射相对维数为一时给出精确结构。
On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action
本文研究了带有环面作用的log Calabi-Yau对的Albanese映射的birational isotrivial性，证明了在特定条件下，Albanese映射的两个一般纤维是birationally等价的。
We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C x C)/G of the "square" of a curve by a finite group G of automorphisms that contains elements not preserving the factors. We study them through the "further quotients" by (C x C)/G' where G' contains G. As a first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp. The main result is a complete description of the Albanese morphism of S through a determined further quotient (C x C)/G' that is an étale cover of the symmetric square of a curve. In particula…
On the Albanese morphism of varieties with Frobenius stable Kodaira dimension zero
本文证明了在正特征下，Frobenius 稳定 Kodaira 维数为零的正规固有簇的 Albanese 态射具有可分、满射和连通纤维等性质，并由此推导出阿贝尔簇上纤维化 Iitaka 次可加性猜想的一个新情形。
Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories
本文证明光滑射影簇的副典范基点轨迹等于其对偶层相对 Albanese 态射的相对基点轨迹，并据此推出 Hilbert 概形及族中导出范畴的半正交不可分解性。
Let $(X, ω)$ be an n-dimensional compact Kähler manifold. Let $D=\sum (1-β_j) Y_j=\sum (1-β_j) [s_j=0]$ a divisor with simple normal crossings with $β_j \in ]0,1[$ such that $-(K_X+D)$ is nef. We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.
On the Existence of Good Minimal Models for Kähler Varieties with Projective Albanese Map
本文研究了凯勒变种在阿尔巴因映射为投影时的良好最小模型存在性，基于KLT对和一般纤维的良好最小模型。
Orbits in stability manifolds and categorical entropy: applications to varieties with finite Albanese morphisms
本文证明具有有限 Albanese 态射的光滑射影簇满足 Gromov--Yomdin 性质，通过轨道逃逸原理和有限平展覆盖归约，并构造正范畴熵自等价，其存在性等价于非一般型。
Let $(X,Δ)$ be a pair. We study how the condition $κ(K_X + Δ)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p > 0$. In particular in characteristic $0$ we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic $p>0$ we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if $X$ is a normal projective threefold in characteristic $p>0$, the coefficients of the components of $Δ$ are $\le 1$ and $-(K_X+Δ)$ is semiample, then the Albanese …
Let $f: X\to Y$ be a surjective morphism of smooth $n$-dimensional projective varieties, with $Y$ of maximal Albanese dimension. Hacon and Pardini studied the structure of $f$ assuming $P_m(X)=P_m(Y)$ for some $m\geq 2$. We extend their result by showing that, under the above assumtions, $f$ is birationally equivalent to a quotient by a finite abelian group. We also study the pluricanonical map of varieties of maximal Albanese dimesnion. The main result is that the linear series $|5K_X|$ incuces the Iitaka model of $X$.
Albanese maps for open algebraic spaces
本研究将Serre关于代数簇到阿贝尔簇泛态射的经典结果推广到开代数空间，证明满足条件的代数空间存在到仿阿贝尔簇的泛态射，给出证明依赖的技术，并分析了曲线与群的情况及非完满基域情形。
Let X be a projective variety over an algebraically closed base field, possibly singular. The aim of this paper is to show that the generalized Albanese variety of Esnault-Srinivas-Viehweg can be computed from one general curve C in X, if the base field is of characteristic 0. We illustrate this by an example, which we also use to unravel some mysterious properties of the Albanese of Esnault-Srinivas-Viehweg.
In this paper we study the cohomology of smooth projective complex surfaces $S$ of general type with invariants $p_g = q = 2$ and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the cohomology of the Albanese variety and a K3 surface $X$ that we call the K3 partner of $S$. Furthermore, we show that in suitable cases we can geometrically construct the K3 partner $X$ and an algebraic correspondence in $S \times X$ that relates the cohomology of $S$ and $X$. Finally, we prove the Tate and Mumford-Tate conjectures for those surfaces $S$ that li…
We consider categories of rational maps to algebraic groups and study existence and construction of universal objects for such categories, using the duality theory of Laumon 1-motives. In particular, we obtain functorial descriptions of the Albanese of a singular projective variety X over an algebraically closed field of characteristic zero, which is the universal regular quotient of a relative Chow group of points on X, and of its dual.
Stability conditions on free abelian quotients
研究光滑射影簇在有限阿贝尔群自由作用下的商簇上的斜率稳定向量丛和Bridgeland稳定性条件，证明了覆盖上的G不变几何稳定性条件与商簇上关于群表示的对偶群不变的几何稳定性条件之间的解析同构，并应用于描述自由阿贝尔商簇稳定性流形中的连通分支。
Let V be a smooth quasi-projective complex surface such that the three first logarithmic plurigenera are equal to 1 and the logarithmic irregularity is equal to 2. We prove that the quasi-Albanese morphism of V is birational and there exists a finite set S such that the quasi-Albanese map is proper over the complement of S in the quasi-Albanese variety A(V) of V. This is a sharp effective version of a classical result of Iitaka.
Let $(X ,x_0)$ be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for $(X ,x_0)$ produces a homomorphism from the abelianization of the $F$-divided fundamental group scheme of $X$ to the $F$-divided fundamental group of the Albanese variety of $X$. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.
A strong counterexample to the log canonical Beauville--Bogomolov decomposition
本研究构造了维数不小于4的对数典范K-平凡簇作为对数典范情形下Beauville--Bogomolov分解的强反例，同时给出相关拟射影簇构造，并证明反典范类nef的对数典范配对的Albanese态射为局部稳定族。
We study normal compact Kähler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on the degree of the Albanese morphism and the number of simple factors of the Albanese variety for rational cohomology tori of general type (hence projective) with rational singularities. Their properties are related to the birational geometry of smooth projective varieties of general type, maximal Albanese dimension, and with vanishing holomorphic Eu…
On surjective morphisms to abelian varieties and a generalization of the Iitaka conjecture
本文研究到阿贝尔簇的满射态射诱导的纤维化，提出并证明了Iitaka猜想在极大Albanese维数基上的推广，预测纤维化Kodaira维数相等。
Let X be a projective variety, possibly singular. A generalized Albanese variety of X was constructed by Esnault, Srinivas and Viehweg over algebraically closed base field as a universal regular quotient of the relative Chow group of 0-cycles by Levine-Weibel. In this paper, we obtain a functorial description of the Albanese of Esnault-Srinivas-Viehweg over a perfect base field, using duality theory of 1-motives with unipotent part.
Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。
We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities on an irregular variety $X$ are related to some natural defined birational invariants of a general fiber $F$ of the Albanese morphism of $X$. As applications, we provide a new lower bound of volumes of irregular threefolds, a sharp lower bound of varieties of maximal Albanese dimension and of general type, and show that the canonical model of such a variety with the minimal volume should be a flat double covers of a principally polarized ab…
Let $(X,B)$ be a complex projective klt pair, and let $f\colon X\to Z$ be a surjective morphism onto a normal projective variety with maximal albanese dimension such that $K_X+B$ is relatively big over $Z$. We show that such pairs have good log minimal models.
Let X be a proper smooth variety over the complex numbers. We consider the generalized Albanese variety Alb(X,Y) of X of modulus Y, which is a higher dimensional analogue of the generalized Jacobian variety with modulus of Rosenlicht-Serre. Note that the divisor Y can have multiplicity, so the algebraic group Alb(X,Y) can have an additive part. The purpose of this paper is to give Hodge theoretic presentations of Alb(X,Y).
On the canonical degree of a Gorenstein minimal threefold of general type
本文证明一般型Gorenstein极小三维流形X的典范映射次数不超过72，当p_g(X)>243时，并刻画了等号成立时X的Albanese纤维的性质。
Generic Vanishing, 1-forms, and Topology of Albanese Maps
本文证明 Abel 簇上可构造层复形的上同调跳跃位点与奇异支集的等式，并用于刻画带零点全纯 1-形式、重证 Albanese 态射相关结果。
Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X,…
We show that the general fibres of the Albanese morphism of a projective special manifold are special as well (a question raised by the first-named author). The main ingredient of the proof is a version (established by Birkar and Chen) with boundary of the famous conjecture of Iitaka on the subadditivity of the Kodaira dimension in an algebraic fibre space. Some consequences of our main result are discussed.
We give a sharp lower bound for the selfintersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the Generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $Vol(X) \geq 2 n! χω_X$ and it is sharp.
We prove a full generalization of the Castelnuovo's free pencil trick. We show its analogies with the Adjoint Theorem; see L. Rizzi, F. Zucconi, Differential forms and quadrics of the canonical image, arXiv:1409.1826 and also Theorem 1.5.1 in G. P. Pirola, F. Zucconi, Variations of the Albanese morphisms, J. Algebraic Geom. 12 (2003), no. 3, 535-572. Moreover we find a new formulation of the Griffiths's infinitesimal Torelli Theorem for smooth projective hypersurfaces using meromorphic $1$-forms.
Effective characterization of ordinary abelian varieties, and beyond
该研究在正特征代数几何领域，证明了满足特定条件的正规真簇的Albanese态射性质及对应Albanese簇的普通性，并给出复阿贝尔簇有效双有理刻画的纯正特征新证明。
Varieties with representable CH_0-group and a question of Colliot-Thélène
本文通过构造一个具有可表示CH_0群但无universal 0-cycle的代数簇，回答了Colliot-Thélène的问题，并利用Benoist和Ottem的反例来证明。
When is the diagonal contractible?
研究光滑射影复簇\(X\)中对角线可收缩的问题，证明其与\(X\)的阿尔巴内塞态射相关条件有关，还给出对角线外为同构的收缩的充要条件，开启对更高次积中对角线收缩的研究。
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…
We prove boundedness of global strong $(δ,n)$-complements for generalized $ε$-log canonical pairs of Fano type. We also prove some partial results towards boundedness of local strong $(δ,n)$-complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong $(δ,n)$-complements implies a conjecture due to M$^{\rm c}$Kernan concerning the singularities of the base of a Mori fibe…
In this article we study the structure of klt projective varieties with nef anticanonical divisor (and more generally, varieties of semi-Fano type), especially the canonical fibrations associated to them. We show that: 1. the Albanese map for such variety is a locally constant fibration (that is, an analytic fibre bundle with connected fibres that $X$ is equal to the product of the universal cover of the Albanese torus by the fibre of the Albanese map quotient by a diagonal action of the fundamenatl group of the Albanese torus); 2. if the smooth locus is simply connected, the MRC fibration of …
Hyperbolicity and fundamental groups of complex quasi-projective varieties (I): Maximal quasi-Albanese dimension by Nevanlinna theory
本文利用 Nevanlinna 理论，对定义在穿孔圆盘小分歧覆盖上、取值于极大拟 Albanese 维数且为对数一般型的复拟射影簇的全纯映射建立大 Picard 型定理，并由此证明广义 Green-Griffiths-Lang 猜想。
Perverse filtration for generalized Kummer varieties of fibered surfaces
本文研究2维群概形纤维化的广义Kummer簇导出的真态射，证明其关联的反常层滤具有乘性。
On the support of admissible subcategories
本文研究光滑射影簇上可容许子范畴的支撑集，通过证明支撑集到阿贝尔簇的态射的纤维无孤立点，得出支撑集不能同构于阿贝尔簇等结论，关键输入是Picard群在无穷小加厚下的提升性质。
In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $φ$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $φ$ is an embedding). This contrasts with the …
This is my PhD Thesis, part of it has published in Acta Mathematica Sinica. In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structures on this family of morphisms.
The classical arithmetic Grothendieck-Riemann-Roch theorem can be applied only to projective morphisms that are smooth over the complex numbers. In this paper we generalize the arithmetic Grothendieck-Riemann-Roch theorem to the case of general projective morphisms between regular arithmetic varieties. To this end we rely on the theory of generalized analytic torsion developed by the authors.
We study the greatest common divisor problem for torus invariant blowing-up morphisms of nonsingular toric Fano varieties. Our main result applies the theory of Okounkov bodies together with an arithmetic form of Cartan's Second Main theorem, which has been established by Ru and Vojta. It also builds on Silverman's geometric concept of greatest common divisor. As a special case of our results, we deduce a bound for the generalized greatest common divisor of pairs of nonzero algebraic numbers.
Over a field of positive characteristic, a semisimple algebraic group $G$ may have some nonreduced parabolic subgroup $P$. In this paper, we study the Schubert and Bott-Samelson-Demazure-Hansen (BSDH) varieties of $G/P$, with $P$ nonreduced, when the base field is perfect. It is shown that in general the Schubert and BSDH varieties of such a $G/P$ are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties are also studied. It is shown that th…
We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective morphisms.
Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary…
Let $X$ be a smooth complex projective variety with nef $\bigwedge^2 T_X$ and $\dim X \geq 3$. We prove that, up to a finite étale cover $\tilde{X} \to X$, the Albanese map $\tilde{X} \to {\rm Alb}(\tilde{X})$ is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety $F$ with nef $\bigwedge^2 T_F$. As a bi-product, we see that either $T_X$ is nef or $X$ is a Fano variety. Moreover we study a contraction of a $K_X$-negative extremal ray $φ: X \to Y$. In particular, we prove that $X$ is isomorphic to the blow-up of a projective space at a point if $φ$ is of birational t…
Projective Representations II. Generalized chain geometries
本文研究广义链几何的射影表示，探讨链映射为直纹面的条件，并确定某类几何的所有双射态射。
Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture
本文引入广义 Hessenberg 簇，证明对称群作用的扩展性，通过双有理几何分析给出 Shareshian-Wachs 猜想及其推广的初等证明，并提出两种计算上同调表示的算法。
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