Council of Elrond

Sage or Dwarf only during the site phase at any Under-deeps site. Tap the sage or Dwarf. Tap the site if it is not already tapped. Sage or Dwarf may not untap until Vein of Arda is stored at a Haven. Cannot be duplicated at a given site.

As I understand it, VoA is played on the given character (even if it's not explicitly stated on the card), so VoA leaves play should the character leave play before it is stored, correct?

As I understand it, VoA is played on the given character (even if it's not explicitly stated on the card)[...]

Because Vein of Arda may be stored at a Haven it implies that the card is possesed/controlled/otherwise associated with entity that is able to be at the Haven.
So yes.

[...] so VoA leaves play should the character leave play before it is stored, correct?

I am not sure wheter I undersand the question.

Situations when Vein of Arda leaves play not in result of leaving a play by given Sage/Dwarf:
- Vein of Arda is stored at Haven (it leaves active play, its text is not in force anymore),
- Vein of Arda is played with Crown of Flowers and Doors of Night or Twilight (on VoA) is successfully played after.

thanks. My second question was: If the character with VoE is, for instance, killed before having stored VoE, VoE is discarded. Now that's clear to me