Bounds on genus and geometric intersections from cylindrical end moduli spaces

Bounds on genus and configurations of embedded surfaces in 4-manifolds

“…(In addition, by using blow-up, we can construct surfaces in 2CP 2 #n(−CP 2 ) that violate the adjunction inequality on c = 3H 1 + 3H 2 − n q=1 E q .) To our knowledge, the adjunction-type inequalities for 2CP 2 #n(−CP 2 ) are only Strle's [19] in previous research. For such 4-manifold, Strle's genus bound is the adjunction inequality for at least one of two disjoint surfaces with positive self-intersection numbers.…”

“…In Strle [24], he showed that the adjunction inequality holds for at least one of disjoint b + surfaces with positive self-intersection numbers. On the other hand, Dai-Ho-Li [1] …”

“…Remark 2.16. Strle [19] and Dai-Ho-Li [1] showed that, when g(Σ) = 0, a sharper result −2 = −χ(Σ) −|c · α| + α 2 holds. Dai-Ho-Li used results in Morgan-Szabó-Taubes [12] to treat this case.…”

“…For example, although the Seiberg-Witten invariant of mCP 2 #n(−CP 2 ) (m, n ≥ 2) vanishes for any spin c structure on it, we can derive the adjunction inequalities for surfaces embedded in this 4-manifold under certain conditions. In addition, we also give an alternative proof of the adjunction inequalities by Strle [24] for configurations of surfaces with positive self-intersection numbers. The proofs of these results are given by studying a b + -parameter family of Seiberg-Witten equations obtained by stretching neighborhoods of several embedded surfaces.…”

“…On the other hand, Strle [24] showed the adjunction inequality for a surface with positive self-intersection number in a 4-manifold with b + = 1 and b 1 = 0 without any assumption for differential geometric structure such as the existence of a metric with positive scalar curvature. For this reason, the Thom conjecture for a general rational homology CP 2 follows as a special case of Strle's result.…”