Remove any dirt or debris from the crack or hole by brushing it away with a stiff paintbrush. If only a soft paintbrush is available, cut the bristles so they are only about 2 inches long.
Clean the area around the hole or crack with rubbing alcohol or acetone to remove any remaining dirt or grease.
Mix the polyester resin according to the manufacturer's directions.
Add marble dust to the resin, following the manufacturer's directions for adding powdered pigment to the resin. If no such directions are available, add the dust in small amounts, mixing between each addition, until the resin is the same color as the marble surrounding the hole or crack.
Spread the resin into the hole or crack using a small putty knife. Make sure the resin is slightly higher than the surface of the surrounding marble. If it is lower, a small dip will still appear on the surface of the marble. If it is even, it may sink slightly, leaving a small dip.
Wait at least 24 hours for the resin to cure completely.
Grind the resin patch using an electric grinder with a fine-grit emery pad until it is smooth and even with the surrounding surface. Grind little by little, taking care not to grind too much or damage the surrounding marble.
Sand the patch and surrounding marble with a fine-grit wet-dry sandpaper.
Wipe away any debris with a damp rag.
Mix marble polishing compound according to the manufacturer's directions
Moisten a felt grinder pad with the polishing compound.
Polish the patch and surrounding area, using the grinder and moistened felt pad until the sheen matches the surrounding marble.