Serve this soup warm or chilled, with plenty of crackers.
Heat oil in a large pot over medium. Add leek and celery and cook, stirring, until softened, about 6 minutes. Add corn kernels, reserved cobs, potato, and 6 cups water. Bring to a boil, then reduce to a simmer and cook until vegetables are very tender, about 30 minutes.
Remove and discard cobs. Working in batches (do not fill jar more than halfway), puree soup in a blender until smooth. Strain through a fine-mesh sieve, pressing on solids to remove as much liquid as possible; discard solids. Whisk in creme fraiche; season with salt and pepper. Thin soup to desired consistency with more water, if needed. Sprinkle with chives and serve, with crackers.