Non Kählerian surfaces with a cycle of rational curves

Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

Calculating the homology and intersection form of a 4-manifold from a trisection diagram

Given a diagram for a trisection of a 4-manifold X, we describe the homology and the intersection form of X in terms of the three subgroups of H1(F;Z) generated by the three sets of curves and the intersection pairing on H1(F;Z). This includes explicit formulas for the second and third homology groups of X as well an algorithm to compute the intersection form. Moreover, we show that all (g;k,0,0)-trisections admit “algebraically trivial” diagrams.

SPLITTING A 4-MANIFOLD WITH INFINITE CYCLIC FUNDAMENTAL GROUP, REVISED

This article is a revised version of the author's earlier paper on a TOP-splitting of a closed connected oriented 4-manifold with infinite cyclic fundamental group. We show that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if it is virtually TOP-split. As a consequence, we see that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if the intersection form is indefinite. This also implies that every closed connected oriented smooth spin 4-manifold with infinite cyclic fundamental group is TOP-split.