Iitaka conjecture $C_{n,m}$ in dimension six

A note on Iitaka's conjecture $C_{3,1}$ in positive characteristics

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 $$κ(X)\geκ(Y) + κ(X_η).$$

关于到阿贝尔簇的满射态射及Iitaka猜想的一个推广

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。

Uniform rational polytopes for Iitaka dimensions

In this paper, we continue to develop the theories on functional pairs and uniform rational polytopes. We show that there is a uniform perturbation for Iitaka dimensions of pseudo-effective lc pairs of fixed dimension with DCC coefficients assuming the non-vanishing conjecture. We also show the existence of uniform rational polytopes for Iitaka dimensions of pseudo-effective lc pairs assuming the non-vanishing conjecture.

Iitaka-Viehweg conjectures C and C++

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.

Iitaka dimension for cycles

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.

关于正特征中反-canonical 分割子在Iitaka猜想中的研究

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。

Iitaka's $C_{n,m}$ conjecture for 3-folds in positive characteristic

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. $$κ(X)\geκ(X_{\overlineη})+κ(Z).$$

关于Iitaka猜想$C_{n,m}$的注记

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} $。

Iitaka-Severi's Conjecture for Complex Threefolds

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$.

On subadditivity of Okounkov bodies for algebraic fiber spaces

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…

On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces

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 …

关于对数典范曲面与三维簇的Iitaka体积

On the Iitaka volumes of log canonical surfaces and threefolds

该研究证明三维及以下对数典范对的Iitaka体积集合在系数集满足降链条件时也满足该条件，刻画了低维特定曲面的Iitaka体积集性质，部分证实了高维klt对的相关猜想。

On subadditivity of the logarithmic Kodaira dimension

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.

Kodaira dimension of algebraic fiber spaces over abelian varieties

In this short article we provide a proof of the Iitaka conjecture for algebraic fiber spaces over abelian varieties.

关于维数至多为4的某些对的锥猜想

On the cone conjecture for certain pairs of dimension at most 4

本文利用极小模型纲领和反正则纤维化，在维数至多为4且Iitaka维数满足特定下界条件下证明了klt Calabi-Yau对的Morrison-Kawamata锥猜想。

Algebraic fiber spaces whose general fibers are of maximal Albanese dimension

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.

On a conjecture of Hacon and McKernan in dimension three

We prove that there exists a universal constant $r_3$ such that if $X$ is a smooth projective threefold over $\mathbb{C}$ with non-negative Kodaira dimension, then the linear system $|r K_X|$ admits a fibration that is birational to the Iitaka fibration as soon as $r \geq r_3$ and sufficiently divisible. This gives an affirmative answer to a conjecture of Hacon and McKernan in the case of threefolds. Viehweg and Zhang have posted a stronger result along these lines using different methods.

Kodaira dimension of algebraic fiber spaces over surfaces

In this short note we prove the Iitaka C_nm conjecture for algebraic fiber spaces over surfaces

奇异度量与 Campana-Peternell 猜想

Singular metrics and a conjecture by Campana and Peternell

本文利用多重伴随丛上的奇异度量理论，证明了 Campana-Peternell 猜想几乎等价于非消失猜想，从而揭示了这两个重要代数几何猜想之间的深刻联系。

三重域上代数纤维空间的柯西维数：第一部分

Kodaira dimension of algebraic fiber spaces over threefolds : Part 1

研究三重域上代数纤维空间的柯西维数行为，证明了Iitaka猜想C_{n,3}的部分情况，包括基域为卡比拉-耶亚三重域的情形。

Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun

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).

The Kobayashi pseudometric on algebraic manifold and a canonical fibration

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…

广义库默尔型超凯勒六维流形的霍奇猜想和泰特猜想

The Hodge and Tate conjectures for hyper-Kähler sixfolds of generalized Kummer type

研究广义库默尔型六维超凯勒簇，证明其霍奇猜想和泰特猜想。

正特征代数纤维空间的数值小平维数

Numerical Kodaira dimension of algebraic fiber spaces in positive characteristic

本文在正特征下证明了代数纤维空间数值小平维数的Nakayama不等式，并建立了相关定理的变体，进而证明了特定条件下Iitaka猜想成立。

Supersingular O'Grady varieties of dimension six

O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supers…

Proof of the Rokhlin's Conjecture on Arnold's surfaces

In this paper we prove that Arnold Surfaces of all real algebraic curves of even degree with non-empty real part are standard (Rokhlin's Conjecture). There is an obvious connection with classification of Arnold Surfaces up to isotopy of S^4 and Hilbert's Sixteen Problem on the arrangements of connected real components of curves. First, we consider some M-curves, i.e curves of a prescribed degree having the greatest possible number of connected real components, and prove that Arnold surfaces of these curves are standard. Afterwards, we exhibit a procedure of modification "perestroika" of these …

Boij-Söderberg锥体中的算术

Arithmetic in the Boij Söderberg Cone

本文通过将猜想转化为Boij-Söderberg锥体中的算术问题，研究了格雷码模的Betti数下界，并在五重和六重情况下证明了新的结果，揭示了Betti表、交换代数和丢番图方程之间的联系。

A note on Grothendieck's standard conjectures of type C and D

Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's conjectures in the case of intersections of quadrics, line…