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Let X be a normal projective variety with klt singularities and LX a strictly nef ℚ-divisor on X. In this paper, we study the singular version of Serrano’s conjecture, i.e. the ampleness of KX + tLX for t ≫ 1. We show that, if X is a ℚ-factorial Gorenstein terminal threefold, then KX + tLX is ample for t ≫ 1 unless X is a weak Calabi–Yau variety (i.e. the canonical divisor KX is a ℚ-torsion and the augmented irregularity q∘(X) vanishes) with LX ⋅ c2(X) = 0. [ABSTRACT FROM AUTHOR]

In this paper, inspired by work of Fano, Morin, and Campana–Flenner, we give a projective classification of varieties of dimension 3 whose general hyperplane sections have negative Kodaira dimension, and we partly extend such a classification to varieties of dimension n⩾4$n\geqslant 4$ whose general surface sections have negative Kodaira dimension. In particular, we prove that a variety of dimension n⩾3$n\geqslant 3$ whose general surface sections have negative Kodaira dimension is birationally equivalent to the product of a general surface section times Pn−2${\mathbb {P}}^{n-2}$ unless (possibly) if the variety is a cubic hypersurface. [ABSTRACT FROM AUTHOR]

We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira–Thurston manifolds. [ABSTRACT FROM AUTHOR]

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than 138. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula. [ABSTRACT FROM AUTHOR]

We explore the relationship between fibrations arising naturally from a surjective morphism to an abelian variety. These fibrations encode geometric information about the morphism. Our study focuses on the interplay of these fibrations and presents several applications. Then we propose a generalization of the Iitaka conjecture which predicts an equality of Kodaira dimension of fibrations, and prove it when the base is a smooth projective variety of maximal Albanese dimension. [ABSTRACT FROM AUTHOR]

We define reflective modular forms on complex balls and use a method of Gritsenko and Hulek to show that some ball quotients of dimensions 3, 4 and 5 are uniruled. We give examples of Hermitian lattices over the rings of integers of imaginary quadratic fields Q (- 1) and Q (- 2) for which the associated ball quotients are uniruled. Our examples include the moduli space of 8 points on P 1 . Moreover, we find that some of their Satake-Baily-Borel compactifications are rationally chain connected modulo certain cusps. [ABSTRACT FROM AUTHOR]

We prove several results on the additivity of the Kodaira dimension under smooth morphisms of smooth projective varieties. [ABSTRACT FROM AUTHOR]

Let Z be a nondegenerate hypersurface in a d-dimensional torus (C*)d defined by a Laurent polynomial f with a d-dimensional Newton polytope P. The subset F(P) ⊂ P consisting of all points in P having integral distance at least 1 to all integral supporting hyperplanes of P is called the Fine interior of P. If F(P) ≠ ∅, we construct a unique projective model Z of Z having at worst canonical singularities and obtain minimal models Z of Z by crepant morphisms b Z → Z. We show that the Kodaira dimension κ = κ(Z) equals min{d - 1, dim F(P)} and the general fibers in the Iitaka fibration of the canonical model Z are nondegenerate (d - 1 - ΰ)-dimensional toric hypersurfaces of Kodaira dimension 0. Using F(P), we obtain a simple combinatorial formula for the intersection number (KZ)d-1. [ABSTRACT FROM AUTHOR]

We study the Noether-Lefschetz locus of the moduli space M of K 3 [ 2 ] -fourfolds with a polarization of degree 2. Following Hassett's work on cubic fourfolds, Debarre, Iliev, and Manivel have shown that the Noether-Lefschetz locus in M is a countable union of special divisors M d , where the discriminant d is a positive integer congruent to 0 , 2 , or 4 modulo 8. We compute the Kodaira dimensions of these special divisors for all but finitely many discriminants; in particular, we show that for d > 224 and for many other small values of d , the space M d is a variety of general type. [ABSTRACT FROM AUTHOR]

We study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma. [ABSTRACT FROM AUTHOR]

We show that the moduli space R ‾ 9 of Prym curves of genus 9 is uniruled. This is the largest genus for which such a result is known to hold. [ABSTRACT FROM AUTHOR]

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: 3 , − 1 , 4 and 4 , λ , − (1 + λ) with − 1 < λ < − 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 of the canonical connection for the almost Kähler structure. Regarding the other two manifolds we compute the Kodaira dimension for certain almost complex structures. Finally, we construct a natural hypercomplex structure providing a twistorial description. [ABSTRACT FROM AUTHOR]

We apply the existence and special properties of Gauduchon metrics to give several applications. The first one is concerned with the implications of algebro-geometric nature under the existence of a Hermitian metric with nonnegative holomorphic sectional curvature. The second one is to show the non-existence of holomorphic sections on Hermitian vector bundles under certain conditions. The third one is to give a restriction on the ∂ ∂ ¯ -closedness of some real (n − 1, n − 1)-forms on compact complex manifolds. [ABSTRACT FROM AUTHOR]

Let X be an elliptic surface over P 1 with κ (X) = 1 , and M = M (c 2) be the moduli scheme of rank-two stable sheaves E on X with (c 1 (E) , c 2 (E)) = (0 , c 2) in Pic (X) × ℤ. We look into defining equations of M at its singularity E , partly because if M admits only canonical singularities, then the Kodaira dimension κ (M) can be calculated. We show the following: (A) E is at worst canonical singularity of M if the restriction of E η to the generic fiber of X has no rank-one subsheaf, and if the number of multiple fibers of X is a few. (B) We obtain that κ (M) = { 1 + dim (M) } / 2 and the Iitaka program of M can be described in purely moduli-theoretic way for c 2 ≫ 0 , when χ (𝒪 X) = 1 , X has just two multiple fibers, and one of its multiplicities equals 2. (C) On the other hand, when E η has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether E is at worst canonical singularity or not. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of generalized Kodaira dimension with multiplier ideal sheaves, and prove the subadditivity of these generalized Kodaira dimensions for certain Kähler fibrations, which was originally obtained for Kodaira dimensions of algebraic fiber spaces by Kawamata and Viehweg. Our method is analytic and based on some new results in recent years. The crucial step in our proof is to prove an L 2 k extension theorem for twisted pluricanonical sections on compact Kähler manifolds. Moreover, we also discuss the relation between two previous optimal L 2 extension theorems with singular weights on weakly pseudoconvex Kähler manifolds. [ABSTRACT FROM AUTHOR]

The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost Kähler manifold and show that in the previous example (and in another one obtained from a Kodaira surface) the Ricci curvature of the almost Kähler metric vanishes for all the members of the family. [ABSTRACT FROM AUTHOR]

In this paper, we prove abundance for 3-folds with nontrivial Albanese maps, over an algebraically closed field of characteristic p > 5. [ABSTRACT FROM AUTHOR]

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra condition. [ABSTRACT FROM AUTHOR]

We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of nonzero degree. [ABSTRACT FROM AUTHOR]

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the (1, 1) curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and classify minimal complex surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are Kähler Calabi-Yau surfaces and Hopf surfaces. [ABSTRACT FROM AUTHOR]

We prove a bound relating the volume of a curve near a cusp in a complex ball quotient X = B/𝛤 to its multiplicity at the cusp. There are a number of consequences: we show that for an n-dimensional toroidal compactification with boundary D, KX + 1 - 𝜆) D is ample for 𝜆 ∈ (0,(n+1)/2𝜋), and in particular that is ample for n ≥ 6. By an independent algebraic argument, we prove that every ball quotient of dimension n ≥ 4 is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture. [ABSTRACT FROM AUTHOR]