The FZ-10 package comes with a lens hood (below left). This lens hood has a filter thread of 72mm for using filters. The right image below shows a 72mm filter mounted in the lens hood.
However, this lens hood has two major problems. First, while the lens hood is wide enough for the wide angle end, it is not deep enough for the telephoto end. In other word, when the lens is zoomed all the way in to 420mm, the angle of view is about 6 degree and the lens hood angle of view is much larger than 6 degree. This means stray light can have a chance to easily strike the lens front element, producing lens flare and/or ghost images. To better shield the lens at the telephoto end, a deeper lens hood is needed. Second, the Panasonic way of suggesting to mount filters on the lens hood could be a bad idea. The FZ-10 lens can focus down to about 2cm at the wide angle end, which is about the same depth from the front glass element of the lens to the filter. Hence, if there are some higher contrast stuffs (e.g., deeper scratch) on the filter, it is likely that the lens may focus on the filter! Moreover, internal reflection among the front glass element, lens barrel and the inner surface of the filter may also produce lens flare. The following image shows a good example. The lens focuses on the dust and scratches of the filter, and you can see clearly the image of the lens barrel (left side of the image). Of course, it was staged to show the problems. In daily shooting this is unlikely to happen.
Given these problems, I would suggest not to mount your filters on the lens hood. Instead, buy an adapter tube and mount our filter on the adapter.
How can we overcome the first problem? There are a number of rubber collapsible lens hoods that may help overcome the problem. Here is a possible solution. Hama as well as several other companies manufacture various sizes collapsible lens hoods. The following images show the Hama's version. It is made from rubber with a metal center tube, actually a thick step ring. The left image below shows the front side with a thread size of 77mm, and the right image shows the back side (to be used on a camera or adapter) with a thread size of 72mm. Thus, if you have a 62mm adapter, you will need a 62-72mm step-up ring in order to screw the lens hood to the adapter. Of course, you will need 77mm filters! There are many combinations of thread sizes available. Try a few different brands will help.
The left image below shows this Hama hood on a PD62 adapter with the help of a 62-72 step-up ring, and the right one shows a 77mm filter mounted on the 77mm thread of the lens hood.
The left image below shows the lens hood extended half way. The collar of the lens hood can be unfolded and the lens hood will extend to its full length as shown in the middle image. The right image shows how deep the lens is and the telephoto end of the lens is completely shielded by the lens hood.
Vignetting seems not a problem with this particular lens hood. I found that even with the lens hood fully extended, vignetting will disappear when the lens is zoomed near to the 2X marking. Therefore, if you like you may use the lens hood in fully extended position, and fold the top portion back when you see vignetting in the EVF or LCD.