Let $M$ be a compact, simply connected smooth manifold and $L\rightarrow M$ a complex line bundle over $M$. Assume there is a continuous path of Kähler structures $(g_t,I_t)\; t\in [0,1]$ on $M$ together with a compatible path of holomorphic structures $J_t$ of the line bundle $L$.
Now we know, that for each fixed $t\in [0,1]$, the space of holomorhic sections $H^0(M,L)_t$ is finite dimensional, hence the linear system $|L|_t := \mathbb{P}(H^0(M,L)_t)$ is a finite dimensional projective space.
Choosing a hermitian metric on $L$, we can equip the space of all smooth sections $\Gamma(X,L)$ with the topology induced by the supremums norm, inducing a topology on the space $\mathbb{P}(\Gamma(X,L))$.
My question now is:
Is the set $\cup_{t\in [0,1]} |L|_t \subseteq \mathbb{P}(\Gamma(X,L))$ compact?