垂青 is a formal, literary verb meaning 'to look upon someone with favor' or 'to show special regard'. It comes from the classical expression '垂青眼' (cast a black-pupil glance — the gaze of favor), and is used in written or high-register speech when a superior or authority figure shows preferential attention. It is a verb-object compound (离合词), though in practice it is rarely split in modern usage. You would see it in contexts like an employer favoring a candidate or a senior scholar taking notice of a junior.