I do not personally keep cories, an yes I do have hard water. Cories prefer soft to moderately hard water (up to 15 dGH). But, it depends on which species you are going to purchase. The smaller cory cats are more finnicky, and delicate as would be expected due to their size. The larger ones will adapt better. For example, both the pepper cory cats and the pygmy cory cats can do well in up to 15 dGH, but the pygmy prefers under 8 dGH.
In short, they can tolerate soft to moderately hard.
But, smaller species generally prefer soft.