The title pretty much says it all – let’s say there is an effect that can destroy objects. If an archer fires an arrow and it intersects this effect, does the arrow get to use the archer’s saves until it reaches its destination, or does it just count as an unattended item at that point?
Carrying, Movement and Exploration defines attended as:
An item attended by a character (being grasped, touched, or worn)...
The arrow, while in flight, is none of those, so it's unattended. Only if the archer uses a magic arrow, does the item receive a save anyway. (Enchanted bows bestow their properties upon their ammunition, but the arrows only count as magical for the purposes of damage reduction).