Place the chicken in the slow cooker and sprinkle with garlic, salt, and pepper. Add the tomatoes, a bit of their oil, the peppers, rosemary, and broth. Cover and cook on low for 5 hours, or until cooked through.
Remove the chicken and vegetables to a platter with a slotted spoon. Pour the remaining liquid into a saucepan and skim off fat from the surface. Bring the juices to a boil over high heat. Stir in the cornstarch mixture and cook, stirring, until slightly thickened. Taste for seasoning, stir in the basil, and pour the sauce over the chicken.