Adjunction inequality for the mixed Heegaard Floer invariant

If ${\Phi }_{M,\mathfrak{s}}\ne 0$ and a connected closed oriented embedded surface $S$ has positive genus and $S^2\ge 0$, thenBlow up $S^2$ points on $S$ and choose the $\pm 1$ exceptional determinant classes which increase the Chern pairing on its proper transform by $S^2$. The Blowup formula for the mixed Heegaard Floer invariant preserves nonvanishing, while the proper transform has square zero. A cut disjoint from it allows the relevant cobordism map to pass through $H{F}^{\pm }(S\times S^1,\mathfrak{s}{|}_{S\times S^1})$. These groups vanish when the Chern pairing exceeds $2g-2$, so the mixed map would then vanish, a contradiction. Reverse the surface orientation for the absolute value. The positive-genus condition is essential: a small nullhomologous sphere has square zero and zero pairing and would violate the displayed inequality.