In this article we study how the birational geometry of a normal projective variety $X$ is influenced by a normal subvariety $A \subset X.$ One of the most basic examples in this context is provided by the following situation. Let $f:X\to Y$ be a surjective holomorphic map with connected fibers between compact connected complex manifolds. It is well known that given a general fiber $A$ of $f$ we have $$ κ(X)\le κ(A)+\dim Y. $$ This article grew out of the realization that this result should be true with $\dim Y$ replaced by the codimension $\cod_X A$ for a pair $(X,A)$ consisting of a normal subvariety $A$ of a compact normal variety $X$ under weak semipositivity conditions on the normal sheaf of $A$ and the weak singularity condition $\cod_A (A\cap\sing X)\ge 2$. We shall now state our main results in the special case of a submanifold $A$ in a projective manifold $X$ and we also simplify the semipositivity notion.
We prove that the image of the pluricanonical representation of a group of bimeromorphic automorphisms of a complex manifold has bounded finite subgroups. As a consequence, we show that the group of bimeromorphic automorphisms of an $n$-dimensional complex manifold whose Kodaira dimension is at least $n-2$, satisfies the Jordan property.
In this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group $G$. It is shown that only for $G = \operatorname{He(3)}, \mathbb Z_3^2$, and only for dimension $\geq 4$ such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension $4$ is given. For the other finite groups a strong structure theorem for rigid quotients is proven.
We conjecture the equality of the numerical and Kodaira dimensions $ν_1^*(X)$ and $κ_1^*(X)$ for the cotangent bundle of compact Kähler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $ν_1^*(X)=κ_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite étale cover of $X$. The proof rests on the Beauville-Bog…
The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact $4$-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: $\mathfrak{r}\mathfrak{r}_{3, -1}$, $\mathfrak{nil}^4$ and $\mathfrak{r}_{4, λ, -(1 + λ)}$ with $-1 < λ< -\frac{1}{2}$. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover we prove Ricci flatness …
For compact complex surfaces (M^4, J) of Kaehler type, it was previously shown that the sign of the Yamabe invariant Y(M) only depends on the Kodaira dimension Kod (M, J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we give a simplified proof of a result that explains why the exclusion of class VII is essential here.
In this note, we apply the semi-ampleness criterion in Lemma 3.1 to prove many classical results in the study of abundance conjecture. As a corollary, we prove abundance for large Kodaira dimension depending only on [BCHM10].
We prove that the variation in a smooth projective family of varieties admitting a good minimal model forms a lower bound for the Kodaira dimension of the base, if the dimension of the base is at most five and its Kodaira dimension is non-negative. This gives an affirmative answer to the conjecture of Kebekus and Kovacs for base spaces of dimension at most five.
We prove that for any smooth complex projective threefold of Kodaira dimension one, the $m$-th pluricanonical map is birational to the Iitaka fibration for every $m\geq5868$ and divisible by $12$.
I propose a few increasingly stronger "superadditivity" conjectures regarding the behavior of Kodaira dimension under morphisms of smooth quasi-projective complex varieties.
Subadditivity of Kodaira dimensions for fibrations of three-folds in positive characteristics
本文在一定的非满性和相对半满性条件下，证明了具有可能奇异几何通类纤维的纤维化的Kodaira维数的次可加性。作为应用，对于特征$p>5$的代数闭域上的光滑三重射影的纤维化$f: X \to Y$，在$Y$是一般型非正则的假设下，我们证明了当一般纤维是光滑的或当$K_{X/Y}$在$Y$上相对较大时Kodaira维数的次可加性.
