Medium Large Transport; Low Wing; 2 Turbojet Eng
Flight Crew Air Transport Pilot (ATP)
ATC Issue All Types; Conflict Ground Conflict; Critical
Human Factors
My FO landed in FAT on RWY 29R and coasted down to the exit where I usually; almost always go; B10. Upon approaching this exit tower told us to turn left at B10; cross RWY 29L and call ground control. I read this back just before the beginning of that turn and we switched controls at that time also. The distance between the runways is very short; and the hold short line; if you do hold short is very soon after exiting RWY 29R. Having made the 90 degree turn I looked left and noticed a GA airplane on very short final crossing over the threshold to RWY 29L. I made a very quick decision to hold short. My FO made a very quick decision and made a call to tower that we were holding short for traffic. The tower acknowledged our call and switched to telling the GA aircraft to land and exit to the left. Shortly after that we were then told to cross RWY 29L and call ground. Later; in the terminal; another Captain came specifically to talk to me about how he witnessed the exact same thing because they were right behind us on B10.I think the reactions of everyone were quickly performed; but adequate to eliminate any actual traffic conflict; but it was just a matter of a second or two. Again; no conflict did occur because of everyone's reaction. As mentioned before it seems neither of us recalls hearing a GA aircraft being cleared to land. ATC; along with the flight crews must maintain situational awareness and be ready to speak up when a potential conflict exists.
Air carrier Captain reported being cleared to cross a runway at FAT where a GA aircraft was on very short final. The Captain elected to hold short of the runway.