Hsueh-Yung Lin
For every compact Kähler manifold $X$ of algebraic dimension $a(X) = \dim X - 1$, we prove that $X$ has arbitrarily small deformations to some projective manifolds.
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Bogomolov decomposition and compact K{ä}hler manifolds of algebraic dimension zero
本文证明了代数维度为零的紧致Kähler流形本质上是Kummer和'简单'流形的积，后者猜想上是双射形辛的。同时证明了四维'严格简单'流形要么是李代数商扭 tori 或者复辛的。
Let $X$ be a compact K{ä}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $π\_1(X)\simeq π\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact K{ä}hler manifolds of algebraic dimension $\dim(X)-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact K{ä}hler threefolds.
We show that any fibration of a 'special' compact K{ä}hler manifold X onto an Abelian variety has no multiple fibre in codimension one. This statement strengthens and extends previous results of Kawamata and Viehweg when $κ$(X) = 0. This also corrects the proof given in [2], 5.3 which was incomplete.
The aim of this paper is to explain the construction by H. Hironaka [H.61] of a holomorphic (in fact "algebraic") family of compact complex manifolds parametrized by $\C$ such for all $s \in \C\setminus \{0\}$ the fiber is projective, but such that the fiber at the origin in non k{ä}hlerian, to mathematicians which are not algebraic geometers. We also explain why it is not possible to make in the same way such an example with fiber at $0$ a simpler example of non k{ä}hlerian Moishezon manifold which is also due to H. Hironaka (see section 5).
We prove that torsion codimension 2 algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension 2 on supersingular abelian varieties over the algebraic closure of finite fields.
We define higher categorical invariants (gerbes) of codimension two algebraic cycles and provide a categorical interpretation of the intersection of divisors on a smooth proper algebraic variety. This generalization of the classical relation between divisors and line bundles furnishes a new perspective on the Bloch-Quillen formula.
Let $Γ$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $ρ:Γ\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(Γ, G)$. It is known from the work of Goldman and Millson that if $Γ$ is the fundamental group of a compact K{ä}hler manifold and $ρ$ has image contained in a compact subgroup then $ρ$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(Γ, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion hol…
We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4.
In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is also true for all smooth varieties $X'$ which are birationally equivalent to $X$. We also prove the analogous result for 1-cycles. As direct corollaries, the Hodge Conjecture holds for smooth rational projective manifolds with dimension less than or equal to five, and, the Generalized Hodge Conjecture holds for smooth rational projective manifolds with dimensi…
Using some relative codimension 1 cycle-space method, we give, following the ideas of D. Popovici [P.13], semicontinuity results for the algebraic dimension in a family a compact complex manifolds parametrized by a disc.
Let $X$ be a non-singular projective threefold over an algebraically closed field of any characteristic, and let $A^2(X)$ be the group of algebraically trivial codimension 2 algebraic cycles on $X$ modulo rational equivalence with coefficients in $\mathbb Q$. Assume $X$ is birationally equivalent to another threefold $X'$ admitting a fibration over an integral curve $C$ whose generic fiber $X'_{\bar η}$, where $\bar η=Spec(\bar {k(C)})$, satisfies the following three conditions: (i) the motive $M(X'_{\bar η})$ is finite-dimensional, (ii) $H^1_{et}(X_{\bar η},\mathbb Q_l)=0$ and (iii) $H^2_{et}…
In his work on birational classification of foliations on projective surfaces, Brunella showed that every regular foliation on a rational surface is algebraically integrable with rational leaves. This led Touzet to conjecture that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. Druel proved this conjecture for the case of weak Fano manifolds. In this paper, we extend this result showing that Touzet's conjecture is true for codimension one foliations on threefolds with nef anti-canonical bundle.
We propose a novel sufficient condition establishing that a piecewise affine variety has the same topology as a variety of the sphere $\mathbb{S}^n$ defined by positively homogeneous $C^1$ functions. This covers the case of $C^1$ varieties in the projective space $\mathbb{P}^n$. We prove that this condition is sufficient in the case of codimension one and arbitrary dimension. We describe an implementation working for homogeneous polynomials in arbitrary dimension and codimension and give experimental evidences that our condition might still be sufficient in codimension greater than one.
Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the Pink-Zilber Conjeture and the Mordel-Lang plus Bogomolov Theorem for curves.
Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild technical hypothesis. We deduce simpler classifications in the special cases of line bundles and vector bundles which are generically local systems. We apply our results to the study of admissible representations of semisimple Lie groups. Our main result gives a new set of constraints on the associated cycles of unipotent representations.
Let $\otimes$ be the map which classifies the tensor product of two line bundles, an extension of this map to the space of all codimension 1 algebraic cycles is constructed. It is proved that this extension cannot exist in codimension greater than 1.
Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that, under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\geq 3$, generalizing an early result of Grothendieck. Next, we show that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ is an algebraic space obtained from a quasi-projective scheme by contracting a curve. This is the first method of descending an Azumaya algebra along a …
On real algebraic realization of round fold maps of codimension $-1$
本文研究了 codimension $-1$ 的 round fold 映射的实代数实现，探讨其在实代数几何中的基础和重要性。
Based on an idea in Hironaka's proof of resolution of singularities, we present an algorithmic smoothness test for algebraic varieties. The test is inherently parallel and does not involve the calculation of codimension-sized minors of the Jacobian matrix of the variety. We also describe a hybrid method which combines the new method with the Jacobian criterion, thus making use of the strengths of both approaches. We have implemented all algorithms in the computer algebra system Singular, and compare the different approaches with respect to timings and memory usage. The test examples originate …
Non-dicritical codimension one foliations on projective spaces of dimension four or higher always have an invariant algebraic hypersurface. The proof relies on a strengthening of a result by Rossi on the algebraization/continuation of analytic subvarieties of projective spaces.
In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of M.Levine-F.Morel claiming that this module has generators in non-negative codimensions. As an application we compute the Algebraic Cobordism ring of a curve. The main tool is Symmetric Operations in Algebraic Cobordism.
Divisorial contractions to codimension three orbits
本文研究连通代数群作用下中心为余维三单连通轨道的等变极值收缩，证明此类收缩均为加权爆破，推广了 Kawakita 的经典结果。
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely t…
Decomposition of the diagonal, intermediate Jacobians, and universal codimension-2 cycles in positive characteristic
本文将复三维簇稳定有理性的两类对角上同调分解障碍推广到正特征有理链连通三维簇，证明相关代数代表元的典范自对偶性，并指出特征>2时带7结点的非常一般四次双立体消解不存在泛余维2闭链类。
We prove that each semialgebraic subset of $\R^n$ of positive codimension can be locally approximated of any order by means of an algebraic set of the same dimension. As a consequence of previous results, algebraic approximation preserving dimension holds also for semianalytic sets.
On the Topology of T-manifolds of Higher Codimension
本文研究了高余维数T-流形的拓扑性质，提出了新的拼接法并构造了最大实代数曲线。
We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matoušek, Sharir and Safernová. We also present a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components …
Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension $2n-2$ or dimension $2n-4$. If the singular set is of dimension $2n-4$, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$ with a meromorphic (rational of degree 1) first integr…
We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism of complex algebraic variety that is injective on the complement of a closed subvariety of codimension at least $2$, is an automorphism.
We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford-Tate conjectures for some classes of algebraic surfaces with $p_g=1$.
On the injective self-maps of algebraic varieties
本文证明了宫西猜想在非奇异簇、曲面及局部完全交集且余维数2处正则的簇上成立，并讨论了诱导非奇异轨迹自同构的实例及孤立奇点情形。
We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space $\mathbb{P}^{n+1}$ of even dimension $n$, passing through the Fermat variety (with $d\neq 3,4,6$). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$. Furthermore, we prove that among all the local Hodge loci associated to a non-linear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1,1,\ldots,1,2)$ attain the minimal possible codimension of their Zariski tangent s…
The 4-Intersection Unprojection Format
本文提出了一种名为4-相交格式的新型反投影方法，用于构造余维6 Gorenstein环，并成功应用于构造加权射影空间中的余维6 Fano三维流形族。
Little is known about the behaviour of the Oka property of a complex manifold with respect to blowing up a submanifold. A manifold is of Class $\mathscr A$ if it is the complement of an algebraic subvariety of codimension at least $2$ in an algebraic manifold that is Zariski-locally isomorphic to $\mathbb C^n$. A manifold of Class $\mathscr A$ is algebraically subelliptic and hence Oka, and a manifold of Class $\mathscr A$ blown up at finitely many points is of Class $\mathscr A$. Our main result is that a manifold of Class $\mathscr A$ blown up along an arbitrary algebraic submanifold (not ne…
We prove an integrality property of the Chern character with values in Chow groups. As a consequence we obtain, for a prime number p, a construction of the p-1 first homological Steenrod operations on Chow groups modulo p and p-primary torsion, over an arbitrary field. We provide applications to the study of correspondences between algebraic varieties.
Deformation of morphisms along leaves of foliations define the tangential foliation on the corresponding space of morphisms. We prove that codimension one fo-liations having a tangential foliation with at least one non-algebraic leaf are transversely homogeneous with structure group determined by the codimension of the non-algebraic leaf in its Zariski closure. As an application, we provide a structure theorem for degree three foliations on $P^3$ .
Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust and applies to a wide range of situations. Consequently, we obtain simple tools for making dimension-based deformation arguments on moduli spaces. Additionally, in a complementary direction we develop the basic properties of codimension for algebraic stacks. One of our goals is to provide a comprehensive toolkit for working transparently with dimension statem…
Pureté de l'approximation forte sur le corps des fonctions d'une courbe algébrique complexe
该研究证明复代数曲线函数域上，半单代数群齐性空间及低次仿射光滑完全交的余维2闭子集补集满足去掉非空有限位集的强逼近性质。
In the present article we discuss an approach to cohomological invariants of algebraic groups over fields of characteristic zero based on the Morava $K$-theories, which are generalized oriented cohomology theories in the sense of Levine--Morel. We show that the second Morava $K$-theory detects the triviality of the Rost invariant and, more generally, relate the triviality of cohomological invariants and the splitting of Morava motives. We describe the Morava $K$-theory of generalized Rost motives, compute the Morava $K$-theory of some affine varieties, and characterize the powers of the fundam…
Using a codimension-$1$ algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety $A$ and showed that the Fourier transform induces a decomposition of the Chow ring $CH^*(A)$. By using a codimension-$2$ algebraic cycle representing the Beauville--Bogomolov class, we give evidence for the existence of a similar decomposition for the Chow ring of hyperKähler varieties deformation equivalent to the Hilbert scheme of length-$2$ subschemes on a K3 surface. We indeed establish the existence of such a decomposition for the…
We prove that there exists a pencil of Enriques surfaces defined over $\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.
On étudie les cycles algébriques de codimension 3 sur les hypersurfaces cubiques lisses de dimension 5. Pour une telle hypersurface, on démontre d'une part que son groupe de Griffiths des cycles de codimension 3 est trivial et d'autre part que l'application d'Abel-Jacobi induit un isomorphisme entre son groupe de Chow des cycles de codimension 3 algébriquement equivalents à zéro et sa jacobienne intermédiaire. ---------- We study 2-cycles of a smooth cubic hypersurface of dimension 5. We show that the Griffiths group of 2-cycles is trivial and the Abel-Jacobi map induces an isomorphism between…
We establish Thom's jet transversality theorem for regular maps from an affine algebraic manifold to an algebraic manifold satisfying a suitable flexibility condition. It can be considered as the algebraic version of Forstnerič's jet transversality theorem for holomorphic maps from a Stein manifold to an Oka manifold. Our jet transversality theorem implies genericity theorems for regular maps of maximal ranks. As an application, it follows that every connected compact locally flexible manifold is the image of a holomorphic submersion from an affine space. We also show that every algebraically …
Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh, Sibony [DS05b], and by Truong [Tru16].Precisely, we give a new proof of the submultiplicativity properties of these degrees and of its birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by Xiao…
We develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension $M\geqslant 4$ and index one over a rationally connected base of dimension at most $\frac12 (M-2)(M-1)$. An estimate for the codimension of the subset of hypersurfaces of a given degree in the projective space with a positive-dimensional singular set is obtained, which is close to the optimal one.
Let $X$ be a projective algebraic manifold and let $CH^r(X)$ be the Chow group of algebraic cycles of codimension $r$ on $X$, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration $\{F^ν\}_{ν\geq 0}$ on $CH^r(X)\otimes {\Bbb Q}$ due to the second author, we construct a space of arithmetic Hodge theoretic invariants $\nabla J^{r,ν}(X)$ and corresponding map $ϕ_{X}^{r,ν} : Gr_{F}^νCH^r(X)\otimes {\Bbb Q} \to \nabla J^{r,ν}(X)$, and determine conditions on $X$ for which the kernel and image of $ϕ_{X}^{r,ν}$ are ``uncountably large''.
We show that a web of codimension at least two and of maximal rank is isomorphic to an algebraic web. This solves a problem first consdered by Chern and Griffiths.
Algebraic ellipticity of quartic double spaces
本文证明特征零代数闭域上所有光滑四次双n重（n≥2）在格罗莫夫意义下代数椭圆，特别回答四次双立体情形的问题4.28，并证明移除余维数至少二的闭子集保持该性质。
Two subanalytic subsets of R^n are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes of order greater than s when r tends to 0. In this paper we prove that every s-equivalence class of a closed semianalytic set contains a semialgebraic representative of the same dimension. In other words any semianalytic set can be locally approximated of any order s by means of a semialgebraic set and hence, by previous results, also by means of an algebraic one (so long as the semianalytic set has codimension at le…
Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result: 1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinski's th…
In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Kodaira's series of papers on semi-regularity (deformations of compact complex submanifolds of codimension 1), deformations of compact complex submanifolds of arbitrary codimensions, and stability of compact complex submanifolds in the context of holomorphic Poisson deformations. We also study simultaneous deformations of holomorphic Poisson structures and holomorphic Poisson submanifolds on a fixed underlying compact complex manifold. In appendices, we present deformations of Poisson closed subscheme…
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