For any Kähler manifold $(X^n,\omega)$ and a holomorphic line bundle $L$ on it, let's define the degree of $L$ to be $\int_X c_1(L)\wedge w^{n-1}$. Now let's take $(X,\omega_1)$ and $(Y,\omega_2)$ two compact Riemann surfaces with $\omega_1,\omega_2$ denoting the respective Kähler forms on them. Now say $L$ is a holomorphic line bundle on $X$, I want to see if deg $L$=deg $\pi_1^*(L)?$ We get \begin{align*} & \int_{X\times Y} c_1(\pi_1^* L)\wedge (\pi_1^*\omega_1+\pi_2^*\omega_2)\\ &=\int_{X\times Y}\pi_1^*c_1 (L)\wedge \pi_2^*\omega_2+\int_{X\times Y}\pi_1^*c_1 (L)\wedge \pi_1^*\omega_1\\ &=\int_{X\times Y}\pi_1^*c_1 (L)\wedge \pi_2^*\omega_2+\int_{X\times Y}\pi_1^*(c_1(L)\wedge\omega_1)\\ &=\int_{X\times Y}\pi_1^*c_1 (L)\wedge \pi_2^*\omega_2+0 \end{align*}
I want to use some form of Fubini to prove my guess but I am not totally sure how to do it. Any help is appreciated.