Thanks for your answer !
But I’m not sure I really get it.
Before posting I tried to modify the page.php to remove the sidebar ( which is not needed when one is on the static front page ), and eventually I ended up removing the part that displays the content to start from scratch. But obviously it means that if I create another post, it won’t have a sidebar.
So do you mean I have to create another $custom.php ( named something like static-page.php ) ? If yes, how do I tell WP to use static-page.php as the front page ?