Which Mongol khagan initiated the invasion of the Song dynasty?

Ögedei Khan initiated the Mongol invasion of the Song dynasty, beginning the long war that eventually brought all of China under Mongol rule.

The campaign began in 1235, during Ögedei's reign as Great Khan. Mongol armies attacked Song territory from several directions, including operations in Sichuan and along the northern approaches. The war was difficult and prolonged because the Song controlled wealthy cities, strong fortifications, major rivers, and an advanced navy.

Ögedei died in 1241, long before the conquest was complete. Möngke Khan later renewed the offensive, while Kublai Khan directed the final phase and proclaimed the Yuan dynasty in 1271. Kublai completed the conquest in 1279 after the Southern Song's defeat at the naval Battle of Yamen.

Genghis Khan had launched the wider Mongol expansion, but he died in 1227, before the organized invasion of the Song began. Kublai was therefore the finisher, not the initiator.