Consider a map $u:(\Sigma,j) \rightarrow (M,\omega)$ where $\Sigma $ is a compact oriented Riemann surface and $M$ is a symplectic manifold.
Show that $\int u^*\omega$ is pure topological, i.e.$\int u^*\omega=<[\Sigma],u^*[\omega]>$ when given either conditions:
$\Sigma$ is closed.
$u(\partial \Sigma)\subset L$ for $L$ a Lagrangian submanifold.
I did not see how to prove using condition 2. Since for the condition 1 part, I prove that it does not depend on cohomology class of $\omega$ by the boundary is empty.
$\int_\Sigma u^*(w+da)=\int u^*\omega+\int_{\Sigma} d(u^*a)=\int u^*\omega+ \int_{\partial \Sigma} u^* a$
But if we have the boundary is mapped to a Lagrangian submanifold, we will only have the symplectic form vanishing on it, but $da$ is an arbitary form, how do we see this?