Related papers: Kodaira Dimension of Fiber Sums along Spheres

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e.,…

In this paper we introduce the notion of relative Kodaira dimension for a symplectic 4-manifold M with a possibly disconnected embedded symplectic surface F. One notable feature is that the sphere components of F have to be discarded, which…

In this note, we verify that the complex Kodaira dimension $\kappa^h$ equals the symplectic Kodaira dimension $\kappa^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many…

In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the…

Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.

Let $(M,\omega)$ be a symplectic 4-manifold of negative Kodaira dimension. Let $C$ be an $\omega$-symplectic curve, $J$-holomorphic for some $J$ tamed by $\omega$. Then $[C]^2$ is bounded below by a constant depending only on $\omega$.…

This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.

We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. In particular, contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of…

This is the second of a series of papers where we study the plurigenera, the Kodaira dimension and the Iitaka dimension on compact almost complex manifolds. By using the pseudoholomorphic pluricanonical map, we define the second version of…

We study Farber's topological complexity for monotone symplectic manifolds. More precisely, we estimate the topological complexity of 4-dimensional spherically monotone manifolds whose Kodaira dimension is not $-\infty$.

We prove that if a closed oriented 4-manifold X fibers over a 2- or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus…

We study the behavior of the Kodaira dimension of algebraic fiber spaces over threefolds. We prove some cases of the Iitaka Conjecture $C_{n,3}$, including certain situations where the base variety is a Calabi--Yau threefold.

In this paper, we will prove subadditivity of Kodaira dimensions for a fibration with possibly singular geometric generic fiber, under certain nefness and relative semi-ampleness conditions. As an application, for a fibration $f: X \to Y$…

We prove several results on the additivity of Kodaira dimension under smooth morphisms of smooth projective varieties.

We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the…

We construct a smooth codimension-one foliation on the five-sphere in which every leaf is a symplectic four-manifold and such that the symplectic structure varies smoothly. Our construction implies the existence of a complete regular…

We discuss the behavior of the Kodaira dimension under smooth morphisms.

We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with $b_1=0$ and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology.…

Given a closed symplectic $4$-manifold $(X,\omega)$, a collection $D$ of embedded symplectic submanifolds satisfying certain normal crossing conditions is called a symplectic divisor. In this paper, we consider the pair $(X,\omega,D)$ with…