The canonical class of a symplectic 4-manifold

Abstract

In this article we present examples of simply connected symplectic 4-manifolds X whose canonical classes are represented by complicated disjoint unions of symplectic submanifolds of X: Theorem. Given finite collections {$g_i$}, {$m_i$}, i=1,...,n, of positive integers, there is a minimal symplectic simply connected 4-manifold X whose canonical class is represented by a disjoint union of embedded symplectic surfaces K ~ $S_{g_1}$,1 « ... « $S_{g_1,m_1}$ « ... « $S_g_n$,1 «... « S{$g_n$,$m_n$} where $S_{g_i},j$ is a surface of genus $g_i$. Furthermore, $c_1^2(X) = c_h(X)$ - (2+ b) where b= $S{i=1}^n m_i$ is the total number of connected components of the symplectic representative of the canonical class.