In this paper we consider the compactness of $\beta$-symplectic critical surfaces in a K\"ahler surface. Let $M$ be a compact K\"ahler surface and $\Sigma_i\subset M$ be a sequence of closed $\beta_i$-symplectic critical surfaces with $\beta_i\to\beta_0\in (0,\infty)$. Suppose the quantity $\int_{ \Sigma_i}\frac{ 1}{ \cos^q\alpha_i}d\mu_i$ (for some $q>4$) and the genus of $\Sigma_{ i}$ are bounded, then there exists a finite set of points ${ \mathcal S}\subset M$ and a subsequence $\Sigma_{ i'}$ that converges uniformly in the $C^l$ topology (for any $l<\infty$) on compact subsets of $M\backslash { \mathcal S}$ to a $\beta_0$-symplectic critical surface $\Sigma\subset M$, each connected component of $\Sigma\setminus { \mathcal S}$ can be extended smoothly across ${ \mathcal S}$.
Categories: math.DG