Direct images of relative pluricanonical bundles

We discuss the local freeness and the numerical semipositivity of direct images of relative pluricanonical bundles for surjective morphisms between smooth projective varieties with connected fibers. We give a desirable semipositivity theorem under the assumption that the geometric generic fiber has a good minimal model.Comment: 17 pages, v2: minor revisions, v3: title changed, revision following referee's comment

The existence of quasiconformal homeomorphism between planes with countable marked points

We consider quasiconformal deformations of $\mathbb{C}\setminus\mathbb{Z}$. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$. In particular, we characterize the closed subsets of $\mathbb{R}$ whose compliments are quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$.Comment: 13 pages, 4 figures. arXiv admin note: text overlap with arXiv:1405.034

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.Comment: 14 pages, a new version of RIMS-1352. Example 2.5 is ne

Koll\'ar-type effective freeness for quasi-log canonical pairs

We prove Koll\'ar-type effective basepoint-free theorems for quasi-log canonical pairs.Comment: 11 pages, v2: minor revision following referee's comment

Fundamental theorems for semi log canonical pairs

We prove that every quasi-projective semi log canonical pair has a quasi-log structure with several good properties. It implies that various vanishing theorems, torsion-free theorem, and the cone and contraction theorem hold for semi log canonical pairs.Comment: 44 pages, v2: Section 6 is new, v3: very minor revisions, v4: minor revisions, v5: revision following referee's report, v6: small mistakes were corrected, reference list was update

Applications of Kawamata's positivity theorem

In this paper we treat some applications of Kawamata's positivity theorem. We get a weak answer to \cite [Section 3]{KeMaMc}. And we investigate the singularities on the target spaces of some morphisms.Comment: 10 page

The indices of log canonical singularities

Let $(P\in X,\Delta)$ be a three dimensional log canonical pair such that $\Delta$ has only standard coefficients and $P$ is a center of log canonical singularities for $(X,\Delta)$. Then we get an effective bound of the indices of these pairs and actually determine all the possible indices. Furthermore, under certain assumptions including the log Minimal Model Program, an effective bound is also obtained in dimension $n\geq 4$.Comment: 25 page

Koll\'ar--Nadel type vanishing theorem

We prove an analytic generalization of Koll\'ar's vanishing theorem, which contains the Nadel vanishing theorem as a special case.Comment: 4 pages, written for AMC201

Semipositivity theorems for moduli problems

We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Koll\'ar. This completes Koll\'ar's projectivity criterion for the moduli spaces of higher-dimensional stable varieties.Comment: 19 pages, v2: very minor revision, v3: major revisions, v4: revision following referee's report, v5: very minor modification

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 $\omega$-sheaves and $\widehat{\omega}$-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.Comment: 18 pages, v2: Corollary 1.5, which is obviously wrong, was removed, v3: title changed, various revisions, v4: minor revisions, v5: minor revisions, v6: very minor revisions, v7: very minor revisions following referee's comments, v8: Theorem 1.9, Lemma 2.8, and Remark 3.8 are new. some mistakes are correcte