Minimal genus smooth embeddings in S2 × S2 and CP2 # nCP2 with n ⩽ 8

Representing nonnegative homology classes of ℂℙ²#𝕟\overline{ℂℙ}² by minimal genus smooth embeddings

“…for those CP 2 #nCP 2 with Fano surface structures, is that only for those n's, could we get complete knowledge on the symplectic cones. Here we find that the single symplectic form coming from the algebraic structure on CP 2 #nCP 2 for any n, shares the function of the whole symplectic cone in [LiL1].…”

“…The pioneer works of Kronheimer and Mrowka [KM2], and Morgan, Szabo and Taubes [MST] on Thom conjecture and its generalization, and the works of T.J. Li and A.K.Liu ([LL1], [LL2]) inspired by [KM2] and Taubes' works [T1]- [T4], advanced the problem from sphere embeddings to minimal genus embeddings. The minimal genus problem for S 2 -bundles over surfaces has been recently solved completely by the author and T.J. Li [LiL2], and the same problem for positive classes in CP 2 #nCP 2 with n ≤ 6 is solved completely in [LiL1] (including partial results for n = 7, 8). In this paper, we give a lower bound for the minimal genus of any nonnegative class of CP 2 #nCP 2 with arbitrary n, and prove that if n ≤ 9, the above lower bound is equal to the minimal genus.…”

“…It was proved in [LiL1] by using Wall's result that for n ≤ 9, any nonnegative homology class can become a reduced class by a diffeomorphism. One of the observations for this paper is that it is still true for n > 9, and we can consider only the reduced classes when dealing with the minimal genus problem.…”

“…The technique in [LiL1] to realize minimal genus embeddings was to use lines in CP 2 or appeal to the result coming from algebraic geometry. The technique in this paper is to use lines and ellitipic curves together, it allows us to give complete results for nonnegative classes in CP 2 #nCP 2 with n ≤ 9 which includes the results of [Li] and [K2] obtained by different methods, and some results for n > 9.…”

“…In [LiL1], the authors calculated the orbits of the positive classes in CP 2 #nCP 2 with n ≤ 8 under diffeomorphisms for minimal genera 1 and 2 which were shown to be finite, and conjectured that the finiteness should be true for positive genera. In this paper, we prove that this conjecture is not only true for n ≤ 8, but also true for n = 9.…”