Add the olive oil to a medium-sized saucepan, and place it over medium-low heat. Add the onion and poblanos and cook until soft, about 5 minutes.
Add the corn kernels and cumin and cook just for a minute or so, to blend the flavors.
Pour in the broth, turn the heat to high, and bring to a boil. Then immediately turn the heat to a simmer, cover, and cook for 15 minutes.
Use an immersion or standing blender to purée the soup until it's almost smooth. I like to blend it just so that it's somewhat chunky. (Please note that when you blend hot liquids in the blender, you have to hold down the lid very, very firmly! If you don't do this, the hot liquid, the vibration, and the steam will force the lid off! Also, use a folded towel to protect your hands from the possibility of hot liquid seeping out of the lid during blending. If this makes you at all nervous, let the soup cool to room temperature, and then blend.)
Add the salt and black pepper and stir to blend. Then either add equal portions of the basil, tomatoes and avocado to serving bowls and ladle the corn soup over it -- or, add all of it directly to the soup and mix it in gently. This soup is delicious served both hot and cold!