if the glass is transparent, you can use an ultraviolet curing glue to repair the break. You clean the area, apply a small dab and then take it out into direct sunlight to "cure" and set in just moments. Opaque glass is harder to repair, since the UV light must penetrate the glass to cure the UV glue. 3M makes one, there are others. The non-commercial glue site "this to that .com" rates glues and can tell you which is best.
Unless your glass figure is made of borosilicate glass (aka Pyrex) there is no sense trying to fix it in a flame. Borosilicate figures, however, can be repaired successfully by the artist.