Abstract
A set of super-commuting vector fields is defined on the super Grassmannians. A characterization of the Jacobian varieties of super curves (super Schottky problem) is established in the following manner: Every finite dimensional integral manifold of these vector fields has a canonical structure of the Jacobian variety of an algebraic super curve, and conversely, the Jacobian variety of an arbitrary algebraic super curve is obtained in this way. The vector fields restricted on the super Grassmannian of index 0|0 give a completely integrable system of partial super differential equations which gives a new supersymmetric generalization of the KP system. Thus every finite-dimensional solution of this new system gives rise to a Jacobian variety of an algebraic super curve. The correspondence between this super Grassmannian and the group of monic super pseudo-differential operators of order zero (the super Sato correspondence) is also established. Table of
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