# Kodaira dimension of fiber sums along spheres

@article{Dorfmeister2010KodairaDO, title={Kodaira dimension of fiber sums along spheres}, author={Josef G. Dorfmeister}, journal={Geometriae Dedicata}, year={2010}, volume={177}, pages={1-25} }

In this note we discuss the effect of the symplectic sum along spheres in symplectic four-manifolds on the Kodaira dimension of the underlying symplectic manifold. We find that the Kodaira dimension is non-decreasing. Moreover, we are able to obtain precise results on the structure of the manifold obtained from the blow down of an embedded symplectic $$-4$$-4-sphere.

## 16 Citations

### Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

- Mathematics
- 2012

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $$S^1$$S1 with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive…

### Stability and existence of surfaces in symplectic 4-manifolds with b +=1

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2018

We establish various stability results for symplectic surfaces in
symplectic 4-manifolds with
b^{+}=1
. These results are then
applied to prove the existence of representatives of…

### Spherical Lagrangians via ball packings and symplectic cutting

- Mathematics
- 2014

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, $$S^{2}$$ or $$\mathbb{RP }^{2}$$, in symplectic manifolds that are rational or…

### Lantern substitution and new symplectic 4-manifolds with ${b_{2}}^{+} = 3$

- Mathematics
- 2012

Motivated by the construction of H. Endo and Y. Gurtas, changing a positive relator in Dehn twist generators of the mapping class group by using lantern substitutions, we show that 4-manifold…

### SMALL EXOTIC 4 –MANIFOLDS AND SYMPLECTIC CALABI-YAU SURFACES VIA GENUS– 3 PENCILS

- Mathematics
- 2021

. In this article, we introduce a strategy to produce exotic rational and elliptic ruled surfaces, and possibly new symplectic Calabi-Yau surfaces, via constructions of symplectic Lefschetz pencils…

### Small symplectic Calabi-Yau surfaces and exotic 4-manifolds via genus-3 pencils

- Mathematics
- 2015

We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then…

### Multisections of Lefschetz fibrations and topology of symplectic 4–manifolds

- Mathematics
- 2016

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various…

### Kodaira Dimension in Low Dimensional Topology

- Art
- 2015

This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].

### Genus-$3$ Lefschetz Fibrations and Exotic $4$-Manifolds with $b_{2}^{+}=3$

- Mathematics
- 2020

We explicitly construct a genus-$3$ Lefschetz fibration over $\mathbb{S}^{2}$ whose total space is $\mathbb{T}^{2}\times \mathbb{S}^{2}\# 6\overline{\mathbb{C} P^{2}}$ using the monodromy of…

## References

SHOWING 1-10 OF 36 REFERENCES

### Minimality of Symplectic Fiber Sums along Spheres

- Mathematics
- 2010

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…

### Kodaira dimension and symplectic sums

- Mathematics
- 2009

Modulo trivial exceptions, we show that 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…

### SYMPLECTIC 4-MANIFOLDS WITH KODAIRA DIMENSION ZERO

- Mathematics
- 2006

We discuss the notion of the Kodaira dimension for symplectic manifolds in dimension 4. In particular, we propose and partially verify Betti number bounds for symplectic 4-manifolds with Kodaira…

### Associativity properties of the symplectic sum

- Mathematics
- 1996

In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diffeomorphic symplectic $4$-manifolds made out of…

### The Kodaira Dimension of Lefschetz Fibrations

- Mathematics
- 2008

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…

### SYMPLECTIC CUTS

- Mathematics
- 2004

. According to McDuﬀ the blow-up operation in symplectic geometry amounts to a removal of an open symplectic ball followed by a col-lapse of some boundary directions. In this paper I describe a…

### The symplectic sum formula for Gromov–Witten invariants

- Mathematics
- 2000

In the symplectic category there is a 'connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula…

### Uniqueness of Symplectic Canonical Class, Surface Cone and Symplectic Cone of 4-Manifolds with B+ = 1

- Mathematics
- 2000

Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b^+=1, we prove that there is a unique symplectic canonical class up to sign, and any real second…

### Singularities and positivity of intersections of J-holomorphic curves

- Mathematics
- 1994

This chapter is devoted to proving some of the main technical results about J-holomorphic curves which make them such a powerful tool when studying the geometry of symplectic 4-manifolds. We begin by…

### Four-Manifolds without Einstein Metrics

- Mathematics
- 1996

It is shown that there are infinitely many compact simply con- nected smooth 4-manifolds which do not admit Einstein metrics, but nev- ertheless satisfy the strict Hitchin-Thorpe inequality 2 !> 3|"…