I am having confusion that, whenever divine beings incarnate on earth, do they also continue to exist in their original abode?
Let us take the example of Narayana, the supreme god. It is said that whole world is just his dream.
It's been said that, only some amsha of Hari came on earth as different avatars (all dasha avatars). Even when the avatar was on earth, Vishnu was still there in Vaikunta.
Sometimes different avatars were there simultaneously. For example, both Krishna and Parashurama attended Yudhishtira's Rajasuya yagna.
Let us take the example of Lord Shiva. I faintly remember a Puranic story, where Shiva used Pashupatastra on his own incarnation. But the astra recognized the incarnation as Shiva himself and returned without harming him.
That solidified the fact that, even when Shiva's incarnation was just amsa of Lord Shiva.
But there is also confusion on, at the end of Dwapara, all Devas and Brahma came to Krishna and spoke to him as:
O You Foundation of Everything, for You there is no longer an obligation to the godly, and the remaining part of the dynasty has virtually been annihilated by this curse of the brahmins . Therefore we ask You whether You intend to leave for Your Supreme Abode and if You please want to continue with us, the protectors of all worlds and their inhabitants, to protect the servants of Vaikunthha.
Reference: Srimadbhagavatam 11.6.26-27
Here they literally urge him to return to Vaikunta and protect them. Now this causes the confusion that Vishnu was not present in Vaikunta?
Also, we have seen numerous other instances of Devas coming to earth as incarnations.
For example, Bhishma was a one of Asta Vasu (named Prabhasa), was cursed to born on earth. It's said that, once he left earthly body, he became Vasu again.
The Vasus are elemental gods, who are responsible for the basic 8 elements, Prabhasa being responsible for the Sky.
So the question is, if the whole soul of Prabhasa was cursed to be born on earth, not just his amsa, then who was responsible for the sky during his time on earth?
Or is it lesser gods or demi gods like Prabhasa-Bhishma who have full incarnations but only supreme gods like Vishnu or Shiva have a partial incarnation?
Pandavas, Karna etc., were clearly called sons of different Devas, not incarnations. But for Bhishma, his incarnation was due to a curse.