What connection does Shah Abbas have with the falcon shown in the painting?

He presented the falcon as a gift to Jahangir

Answer:  He presented the falcon as a gift to Jahangir

The painting Falcon Perched on a Bird Rest by Ustad Mansur, Nadir ul Asr, a title received from Jahangir, is in the collection of Cleveland Museum of Art, Ohio, USA. Jahangir had fine falcons brought to his collection, and as a keen connoisseur, he had them painted. These images were included in his official biography Jahangirnama. There is an interesting episode described by him of a falcon brought as a present from Persian emperor Shah Abbas. This is to do with the falcon, which was mauled by a cat, resulting in its death, and the Emperor desired his painters to paint the dead falcon, preserving its memory for posterity. The painting Falcon Perched on a Bird Rest (1615), is one of the many paintings that were painted by the Mughal artist, Ustad Mansur.