The daughter of a Swedish mother (Tamara Urtahl, a Swede) and of a wealthy St. Petersburg manufacturer of Muslim heritage and patron of the arts who later became a free-thinker (Levko Zheverzheiev), Geva grew up in a vast 18th century mansion which included a private museum. Geva described her mother as beautiful but selfish, frequently unfaithful to her husband. Geva's parents were unable to marry until their daughter was six years old.

Studying ballet was the passion of her childhood, though there were many obstacles before she eventually was able to devote herself to it. Geva met George Balanchine at the Maryinsky Ballet School and married him in 1923 when she was 16. (The marriage was dissolved in 1926.)

The couple joined Diaghilev's Ballet Russes, where Balanchine began to establish a name for himself in Europe. Geva ultimately was dissatisfied with the limited opportunities in the company and joined Balieff's touring 'Chauve-Souris' troupe, which performed in New York in 1927. She and Balanchine divorced but remained on friendly terms. Later, Geva had several huge successes on Broadway, most notably dancing in Richard Rodgers' "Slaughter on 10th Avenue" ballet from 'On Your Toes' (1936) with George Balanchine providing the legendary choreography.