**Core Concept**
Post-transplant recurrence of diseases in the kidney is a significant concern in renal transplantation. The recurrence of diseases like focal segmental glomerulosclerosis (FSGS), membranous nephropathy, and membranoproliferative glomerulonephritis (MPGN) can have a significant impact on graft survival and patient outcomes.

**Why the Correct Answer is Right**
The disease that does not recur in the kidney after renal transplant is **polyoma BK virus nephropathy**. This is because polyoma BK virus nephropathy is caused by the reactivation of a latent virus in the allograft, which is a result of immune suppression, and not a recurrence of the original disease process. The virus infects the renal tubular cells and causes damage, leading to nephropathy. Since the virus is not a recurrence of the original disease, it does not recur in the same way as other diseases.

**Why Each Wrong Option is Incorrect**
**Option A:** FSGS is known to recur in the graft in up to 50% of cases, making it a significant concern in renal transplantation.
**Option B:** Membranous nephropathy can also recur in the graft, although the risk is lower than FSGS.
**Option C:** MPGN is another disease that can recur in the graft, and it is associated with a significant risk of graft failure.

**Clinical Pearl / High-Yield Fact**
It's essential to note that the risk of disease recurrence in the graft can be reduced by optimizing immunosuppressive therapy, reducing the intensity of immune suppression, and using antiviral prophylaxis for polyoma BK virus.

**Correct Answer:** D. Polyoma BK virus nephropathy.