= Adjunction inequality for the mixed Heegaard Floer invariant
{c}
If $\Phi_{M,\mathfrak s}\ne0$ and a connected closed oriented embedded surface $S$ has positive genus and $S^2\ge0$, then
$$
|\langle c_1(\mathfrak s),[S]\rangle|+S^2\le2g(S)-2.
$$
Blow up $S^2$ points on $S$ and choose the $\pm1$ 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 $HF^\pm(S\times S^1,\mathfrak s|_{S\times S^1})$. 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.
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