In "The Great Gatsby" what impression of Gatsby is suggested by the phrase "come out to determine what share was his of our local heavens"? F....

When Nick Carraway sees Mr. Gatsby for the first time, he spots Gatsby

Something in his leisurely movements and the secure position of his feet upon the lawn suggested that it was Mr. Gatsby himself, come out to determine what share was his of our local heavens.

With Gatsby, the imagery of the moon and the heavens is prevalent throughout the novel.  As Nick states later in the novel,

He was a son of God--a phrase which, if it means anything, means just that--and he must be about his Father's Business (104)

Jay Gatsby, the "Great Gatsby" is portrayed by Fitzgerald as a religious entity in several scenes with images of the moon radiating and the heavens looking with favor upon him.  For instance, after the car accident involving Myrtle Wilson and Daisy, Nick encounters Gatsby outside the Buchanan house in his "pink suit under the moon."

Regarding the godly image of the quote under question, it is interesting that this son of God has not come out to determine his part of the universe.  Instead, he seeks what is his "share."  This word suggests the flaws of Gatsby and his ethereal American Dream in which he envisions the moon radiating favor upon him while he feels that there is a white ladder he can climb to a secret place above the trees.

A flawed Chiristlike figure built by the excessive materialism of the Jazz Age, Gatsby gives parties to which Nick feels he is "the only one invited."  The biblical allusion from Matthew of the wedding feast in which many came and were miraculously fed cannot be missed as are other such allusions.  While the Christ-like Gatsby saves Daisy and Tom from the law, he does not, however, save them from their sins.  In the final chapter, Jay Gatsby carries his air mattress much like Christ carrying his cross; Gatsby is shot in the pool and becomes the sacrificial lamb, a tragic figure who represents the tragedy of the flawed materialistic American Dream. Indeed, there was only a small share of the universe that Jay Gatby had.