Markus Wessler
We give a proof of Iitaka's Conjecture C_{2,1} using only elementary methods from algebraic geometry. The main point is that, given a non-isotrivial and relatively minimal family f : X \to B, where X is a surface and B is a curve, both smooth and projective, we show that the direct image of the relatively canonical sheaf has strictly positive degree.
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We prove that the Iitaka conjecture $C_{n,m}$ for algebraic fibre spaces holds up to dimension 6, that is, when $n\le 6$.
Let X be a compact Kähler manifold whose universal covering is $\mathbb C^n$. A conjecture of Iitaka claims that X is a torus, up to finite étale cover. We prove this conjecture in various cases in dimension four. We also show that in the projective case Iitaka's conjecture is a consequence of the non-vanishing conjecture.
We prove that the log Iitaka conjecture holds for log canonical fibrations when log canonical divisor of a sufficiently general fiber is abundant.
We study the relationship between Iitaka fibrations and the conjecture on the existence of complements, assuming the good minimal model conjecture. In one direction, we show that the conjecture on the existence of complements implies the effective log Iitaka fibration conjecture. As a consequence, the effective log Iitaka fibration conjecture holds in dimension $3$. In the other direction, for any Calabi-Yau type variety $X$ such that $-K_X$ is nef, we show that $X$ has an $n$-complement for some universal constant $n$ depending only on the dimension of $X$ and two natural invariants of a gene…
We prove Iitaka's $C_{n,m}$ conjecture for $3$-folds over the algebraic closure of finite fields. Along the way we prove some results on the birational geometry of log surfaces over nonclosed fields and apply these to existence of relative good minimal models of $3$-folds.
Generalized Nonvanishing Conjecture and Iitaka Conjecture
本文证明了对于κ>0或q>0的三维簇成立广义非消失猜想，并由此推导出饭高猜想$C_{n,m}$在n=7及特定条件下成立，且$C^-_{n,m}$在n≤7时成立。
Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_η$ has big canonical divisor $K_{X_η}$, then $$κ(X)\geκ(Y) + κ(X_η).$$
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。
Given a fibre space $X/S$ with the generic geometric fibre of Kodaira dimension $\geq 0$, we shall construct a variety $Y$ ramified over $X$ along such a horizontal hyperplane with respect to $X/S$ that Kollár and Kawamata had proved Viehweg conjecture for $Y/S$ with the generic geometric fibre of general type or of the abundant canonical invertible sheaf where Viehweg dimensions of $X/S$ and $Y/S$ are equal, respectively. We shall show that Viehweg dimension of $X/S$ is not greater than that of $Y/S$ by Mochizuki's Galois theory.
We define the Iitaka dimension of a numerical cycle class and develop its theory. We conjecture that the Iitaka dimension is integer-valued, and give some evidence in this direction. We focus on two cases of geometric interest: Schubert cycles on Grassmannians and cycles contracted by morphisms.
On the Iitaka conjecture for anticanonical divisors in positive characteristic
本文研究正特征下纤维化中的Iitaka型不等式，证明在三重或曲线目标时，若一般纤维正则且诱导对强F-正则，则成立。同时给出特征2和3下非正常纤维的反例。
The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new app…
In this paper, we prove that for a fibration $f:X\to Z$ from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ with $\mathrm{char} k =p >5$, if the geometric generic fiber $X_{\overlineη}$ is smooth, then subadditivity of Kodaira dimensions holds, i.e. $$κ(X)\geκ(X_{\overlineη})+κ(Z).$$
Note on Iitaka Conjecture $C_{n,m}$
本文在Albanese维数$ α(Y) \geq m-2 $的条件下，证明了Iitaka猜想$ C_{n,m} $。
We prove the following generalization of Severi's Theorem: Let $X$ be a fixed complex variety. Then there exist, up to birational equivalence, only finitely many complex varieties $Y$ of general type of dimension at most three which admit a dominant rational map $f$ from $X$ to Y$.
Let $(X,D)$ be a log smooth pair of dimension $n$, where $D$ is a reduced effective divisor such that the log canonical divisor $K_X + D$ is pseudo-effective. Let $G$ be a connected algebraic subgroup of $\mathrm{Aut}(X,D)$. We show that $G$ is a semi-abelian variety of dimension $\le \min\{n-\barκ(V), n\}$ with $V := X\setminus D$. In the dimension two, Shigeru Iitaka claimed in his 1979 Osaka J. Math. paper that $\dim G\le \bar{q}(V)$ for a log smooth surface pair with $\barκ(V) = 0$ and $\bar{p}_g(V) = 1$. We (re)prove and generalize this classical result for all surfaces with $\barκ=0$ wit…
In this short article we provide a proof of the Iitaka conjecture for algebraic fiber spaces over abelian varieties.
By applying the positivity theorem of direct images and a pluricanonical version of the structure theorem on the cohomology jumping loci à la Green-Lazarsfeld-Simpson, we show that the klt Kähler version of the Iitaka conjecture $C_{n,m}$ (Ueno, 1975) for $f:X\to Y$ (surjective morphism between compact Kähler manifolds with connected general fibre) holds true when the determinant of the direct image of some power of the relative canonical bundle is big on $Y$ or when $Y$ is a complex torus. These generalize the corresponding results of Viehweg (1983) and of Cao-P\u aun (2017) respectively. We …
The main purpose of this paper is to prove the Iitaka conjecture $C_{n,m}$ on the assumption that the sufficiently general fibers have maximal Albanese dimension.
Dynamical Iitaka theory on Fano contractions
本文利用分歧除子的动力 Iitaka 纤维化，建立 Mori 纤维空间上满射自同态的结构定理，并证明射影丛上的 Kawaguchi-Silverman 猜想。
On quasi-Albanese maps
本文详细讨论了Iitaka的拟Albanese映射理论，并对对数Kodaira维数为零的簇的Kawamata定理给出了详细证明，指出该理论是Deligne混合Hodge理论的应用。
In this short note we prove the Iitaka C_nm conjecture for algebraic fiber spaces over surfaces
We reduce Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension to a special case of the generalized abundance conjecture by establishing an Iitaka type inequality for Nakayama's numerical Kodaira dimension. Our proof heavily depends on Nakayama's theory of $ω$-sheaves and $\widehatω$-sheaves. As an application, we prove the subadditivity of the logarithmic Kodaira dimension for affine varieties by using the minimal model program for projective klt pairs with big boundary divisor.
Given a compact complex manifold X of dimension n, we define a bimeromorphic invariant $κ_+(X)$ as the maximum p for which there is a saturated line subsheaf L of the sheaf of holomorphic p forms whose Kodaira dimension $κ(L)$ equals p. We call X special if $κ_+(X)=0$. We give some evidence that this condition characterizes X to have identically vanishing Kobayashi pseudometric. We use the well-known construction of F. Campana to give, for each projective X a canonical fibration $f: X\to Y$ holomorphic outside a proper subvariety of Y and whose general fibers are special. We show that the inhe…
We present a simplified proof for a recent theorem by Junyan Cao and Mihai Paun, confirming a special case of Iitaka's conjecture: if $f \colon X\to Y$ is an algebraic fiber space, and if the Albanese mapping of $Y$ is generically finite over its image, then we have the inequality of Kodaira dimensions $κ(X)\geq κ(Y)+κ(F)$, where $F$ denotes a general fiber of $f$. We include a detailed survey of the main algebraic and analytic techniques, especially the construction of singular hermitian metrics on pushforwards of adjoint bundles (due to Berndtsson, Paun, and Takayama).
Let f:X->Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map of relative dimension no more than three, then X has a good minimal model.
The algebraic Sato-Tate conjecture was initially introduced by Serre and then discussed by Banaszak and Kedlaya. This note shows that the Mumford-Tate conjecture for an abelian variety A implies the algebraic Sato-Tate conjecture for A. The relevance of this result lies mainly in the fact that the list of known cases of the Mumford-Tate conjecture was up to now a lot longer than the list of known cases of the algebraic Sato-Tate conjecture.
An approach to the abundance conjecture for Kähler varieties via algebraic reduction
本文通过归纳法建立Kähler变种丰富性猜想的策略，利用代数减少纤维化将问题转化为射影变种的丰富性猜想，特别在四维情况下获得了一些非代数或轨型为零的Kähler四重的丰富性猜想案例。
We consider the Zariski-Lipman Conjecture on free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several classes of affine and projective surfaces.
The conjecture called algebraic Montgomery-Yang problem is still open for rational $\mathbb{Q}$-homology projective planes with cyclic quotient singularities having ample canonical divisor. All known such surfaces have a special birational behavior called a cascade. In this note, we establish algebraic Montgomery-Yang problem assuming the cascade conjecture, which claims that every rational $\mathbb{Q}$-homology projective planes with quotient singularities having ample canonical divisor admits a cascade.
This paper has been withdrawn by the author, due an error in the proof of Proposion 2.13.
In this note we are going to consider a smooth projective surface equipped with an involution and study the action of the involution at the level of Chow group of zero cycles.
In the present paper, we focus on a weighted version of the Bounded Negativity Conjecture which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a global constant. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.
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。
Conjecture I for unirational algebraic groups over imperfect fields
本文基于不完美域上代数群的最新结构进展，证明了Serre猜想I的推广及相关结果，主要贡献是证明单理代数群的第一Galois上同调集在Kato意义下域的上同调维数至多1时为平凡。
Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H \subseteq G$ an observable subgroup normalized by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 70's that the invariant algebra $k[X]^H$ is finitely generated. We prove the conjecture for 1) subgroups of $\mathrm{SL}_n(k)$ closed under left (or right) Borel action and for 2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups…
Informed by the Bloch-Beilinson conjectures, Voisin has made a conjecture about $0$-cycles on self-products of Calabi-Yau varieties. In this note, we consider variant versions of Voisin's conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.
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…
We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties.
We propose a non-commutative generalization of Beilinson's Conjecture on the regulator map from algebraic K-theory to Deligne cohomology of algebraic varieties over Q. We also check a baby case of the generalized conjecture, namely, the case of finite-dimensional associative algebras.
Non-abelian Hodge theory for non-proper varieties and the linear Shafarevich conjecture
本文综述了非交换Hodge理论在非 proper 代数变种中的最新进展，并利用这些工具构造代数Shafarevich映射，证明了对于任何代数变种的线性Shafarevich猜想版本。
We give a theorem of Leray-Hirsch type for Chow groups and use it to study the Hogde and Grothendieck's standard conjectures for algebraic fiber bundles of Leray-Hirsch type. Morevoer, the Hodge conjecture for product varieties will also be considered.
Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partially address the case in which the gonality of C is computed by infinitely many pencils.
De Rham logarithmic classes and Tate conjecture
本文定义了德雷哈姆对数类，证明了代数簇的德雷哈姆类为对数类，并推导出有限型数域上光滑射影簇的塔特猜想。
Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture
本文提出一个符号解析框架，通过精细不变量厄米特谱指纹的消失性来判定绝对霍奇类，从而为霍奇猜想提供新的统一判据与解决路径。
We focus on Voisin's conjecture on 0-cycles on the self-product of surfaces of geometric genus one, which arises in the context of the Bloch-Beilinson filtration conjecture. We verify this conjecture for the family of Todorov surfaces of type $(2,12)$, giving an explicit description of this family as quotient surfaces of the complete intersection of four quadrics in $\mathbb{P}^{6}$. We give some motivic applications.
The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. We show that the polynomial GGL conjecture for a generic projective hypersurface of degree $\mathrm{…
Motivated by the Bloch-Beilinson conjectures, Voisin has formulated a conjecture about 0-cycles on self-products of surfaces of geometric genus one. We verify Voisin's conjecture for the family of Todorov surfaces with $K^2=2$ and fundamental group $\mathbb{Z}/2\mathbb{Z}$. As a by-product, we prove that certain Todorov surfaces have finite-dimensional motive.
Syzygies of algebraic varieties through symmetric products of algebraic curves
本文综述了利用代数曲线对称积的几何方法证明的关于代数簇合冲的若干近期结果，包括亏格猜想、割线簇与切发展曲面的合冲、格林猜想及渐近合冲。
The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.
Motivated by the Beauville-Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperkähler fourfolds.
We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi-Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.
The subject of the present paper is Grothendieck's Lefschetz standard conjecture $B(X)$. Our main result is that, if $X$ is a projective smooth variety of dimension $n$ and the conjecture $B({\cal Y})$ holds for the generic fibre ${\cal Y}$ (of dimension $n-1$ over the field $k(t)$) of a suitable Lefschetz fibration of $X$, then the operator $Λ_X-p^{n+1}_X$ is algebraic. If in addition $p^{n+1}_X$ is algebraic, then $B(X)$ is settled. Along the way we establish the algebraicity of the K\" unneth projectors $π^i_X$ for $i\neq n-1, n, n+1$ under the above hypotheses.
Using Balmer--Favi's generalized idempotents, we establish the telescope conjecture for many algebraic stacks. Along the way, we classify the thick tensor ideals of perfect complexes of stacks.
We show how to deduce the standard sign conjecture (a weakening of the Künneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur's conjectures plus a bit more). We also indicate what is known (to us) about these statements.
An introduction to the algebraic geometry of the Putman-Wieland conjecture
本文以代数几何视角重释 Putman-Wieland 猜想的既有结果，构造 Jacobian 含高维等平凡同源因子的折纸曲线族，并证明其超椭圆类比失败。
Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties
本文探讨复代数簇的双曲性、拓扑结构及沙法列夫猜想等核心问题，通过非阿贝尔霍奇理论和基本群表示等方法，揭示这些领域间的深刻联系与最新进展。
We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain "higher Todd genera" are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the "strong Novikov Conjecture" for the fundamental group, which is known to be correct for many groups …
We reduce the Openness Conjecture of Demailly and Kollár on the singularities of plurisubharmonic functions to a purely algebraic statement.
Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the Generalized Jacobian conjecture in the field of real numbers, which implies the Generalized complex Jacobian conjecture. Also, for the strong real Jacobian conjecture, we present a newly sufficient and necessary condition.
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