In large pot, heat oil over medium-high heat. Add onion, leek and green onions. Reduce heat to medium and cook, stirring often, until vegetables are softened, about 10 minutes.
Add corn, potato, broth, basil and mint. Bring to a gentle boil over medium-high heat. Reduce heat to medium-low, cover and simmer until potato is tender and liquid has reduced to just cover ingredients, about 30 minutes.
Purée mixture using an immersion blender or food processor, until almost smooth but still a little chunky. Season with salt and pepper.