The history of Veneto is fascinating. It has, relatively recently, been part of both Austria and France and was finally returned to Italy in 1866.
There are 28 villages where you can ski in Veneto but the vast majortity are a handful of lifts and can't be considered resorts. This accolade essentially belongs to Cortina D'Ampezzo with 120km of pistes and 35 ski lifts although Arabba with 62km of pistes and 20 lifts is highly respected by the locals.
Cortina is a couple of hours from Venice and would be ideal for a two centre holiday.