Peah, Chapter 2, Mishnah 7. When the Torah teaches us the mitzvah of peah, it links it to the act of harvesting, as it says: "U'vekutzrechem es ketzir artzechem lo sechaleh pe'as sadecha liktzor" - when you harvest the harvest of your land, do not harvest the corner of your field. We learn from this that the obligation of peah depends on the act of harvesting.
This principle is illustrated in our Mishnah:
"Sadeh shekatzruha goyim, ketzaruha listim, kirsemuha nemalim, shevarasah haruach o behemah - peturah" - a field that was harvested by non-Jews, harvested by robbers, gnawed down by ants, or broken by the wind or by an animal, is exempt. These are the cases:
"Shekatzruha goyim" - the harvesting was done by non-Jews, and afterward the cut sheaves were sold to a Jew, or the non-Jews themselves converted after they harvested the grain.
"Ketzaruha listim" - the owner of the field is not the one who harvested; rather, robbers came and harvested the grain, and the grain was left lying on the ground.
"Kirsemuha nemalim" - the grain was eaten and cut down by ants, which attacked the stalks so that they fell.
"Shevarasah haruach o behemah" - the stalks were broken by the wind or by an animal.
In each of these cases the field is exempt from giving peah, because it does not meet the condition of "u'vekutzrechem" - that the Jew, the owner of the field, harvests the grain himself. If he harvested it himself, he is obligated; if he did not harvest it himself, he is exempt.
He harvested half and robbers harvested half:
The owner of the field harvested half of it, and before he managed to harvest the second half robbers came and harvested it - the field is exempt, and the entire field is included in the exemption, even the portion that the owner harvested himself.
The reason: "shechovas hapeah bakamah" - the obligation to give peah takes effect in the standing grain, at the end of the harvesting. And since there, in that final portion, the harvesting was done by someone else and not by the owner of the field, there is no fulfillment of "u'vekutzrechem" here, and therefore the entire field is exempt from giving peah.
In summary: in this Mishnah we learned that the obligation of peah depends on the harvesting being done by the owner of the field himself, as the verse says, "u'vekutzrechem." A field harvested by non-Jews, by robbers, by ants, by the wind or by an animal is exempt. And even one who harvested half his field and then robbers came and harvested the remaining half is exempt from everything, because the obligation of peah takes effect in the standing grain, in the grain that remains at the end of the harvesting.