The Eye of Magnus (quest)

After retrieving the Staff of Magnus from Labyrinthian, you come back to find that the situation has gotten worse.

Use the Staff of Magnus to enter the College of Winterhold
When you reach the edge of the barrier engulfing the college, Tolfdir will fill you in on what has happened while you were gone and tell you to use the staff on the barrier to bring it down. Draw the staff and just blast the barrier with it and it should fall without a fuss.

Reach the Hall of Elements
Once the barrier is down, go straight through the courtyard and into the Hall of Elements. You will find Ancano casting magic at the Eye.

Defeat Ancano
When you find Ancano, Tolfdir will walk up and start trying to reverse the damage by casting a spell at the Eye. At this point, Ancano will turn aggressive and attack. When he attacks, the Eye will open up and the loose panels will start orbiting the centre. While the Eye is open, Ancano cannot be damaged. Use the staff on the Eye until it closes and begin blasting away at Ancano. Tolfdir will be assisting in this fight, but do not expect much from him. Occasionally, Magical Anomalies will spawn near the Eye, so make sure that you are keeping Ancano between yourself and the Eye and they will attack him. The Anomalies will drop Soul Gems that can be used to recharge the staff if it empties.

Talk to Tolfdir
When Ancano dies, Tolfdir will walk up to you and begin speaking. Talk with him and wait for him to finish speaking and Quaranir will show up.

Speak with Quaranir
Walk up to Quaranir and speak with him. He will congratulate you on defeating Ancano and declare you the new Arch-Mage. Two other members of the Order then teleport in and the three of them will teleport the Eye out for safekeeping.

Now officially the new Arch-Mage the Dragonborn receives the Arch-Mage's Quarters Key (now everything here belongs to the Dragonborn), and rather fetching Archmage's Robes from Tolfdir.

Das Auge von Magnus (Aufgabe)