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We compute the complexity, z-complexity, and support varieties of the (thick)Kac modules for the Lie superalgebras of type P. We also show the complexityand the z-complexity have geometric interpretations in terms of support andassociated varieties; these results are in agreement with formulas previouslydiscovered for other classes of Lie superalgebras.  Our main technical tool is a recursive algorithm for constructing projectiveresolutions for the Kac modules. The indecomposable projective summands whichappear in a given degree of the resolution are explicitly described using thecombinatorics of weight diagrams. Surprisingly, the number of indecomposablesummands in each degree can be computed exactly: we give an explicit formulafor the corresponding generating function.

Complexity and Support Varieties for Type P Lie Superalgebras