One of C.Taubes' theorems says that for a symplectic 4-manifold with $b_2^+>1$, $\mathrm{Gr}(e)=0$ if $C_1(K).e-e.e\neq0$. This theorem is proved using Seiberg-Witten theory. And my question is: can one prove this gauge-theory-freely?

One of C.Taubes' theorems says that for a symplectic 4-manifold $X$ with $b^2_+>1$ (where $b^2_+$ denotes the dimension of a maximal positive-definite subspace of $H^2(X;\mathbb R)$ under the intersection form), $\mathrm{Gr}(e)=0$ if $c_1(K)\cdot e-e\cdot e\neq0$. Here $Gr(e)\in\mathbb Z$ is a particular count of $J$-holomorphic curves in $X$ which represent the class $e\in H_2(X;\mathbb Z)$, and $K^{-1}$ denotes the canonical bundle. This theorem is proved using Seiberg-Witten theory. Can one prove this gauge-theory-freely?

One of C.Taubes' theorem says that for a symplectic 4-manifold with b_2^+>1, Gr(e)=0 if C_1(K).e-e.e\neq0. This theorem is proved using Seiberg-Witten theory. And my question is: can one prove this gauge-theory-freely?

One of C.Taubes' theorems says that for a symplectic 4-manifold with $b_2^+>1$, $\mathrm{Gr}(e)=0$ if $C_1(K).e-e.e\neq0$. This theorem is proved using Seiberg-Witten theory. And my question is: can one prove this gauge-theory-freely?