# Classification of superpotentials

## TL;DR

This paper extends the classification of superpotentials related to scalar curvature in cohomogeneity one Ricci-flat equations, exploring cases outside previous convex hull constraints and linking to Calabi-Yau conditions.

## Contribution

It broadens the classification framework by analyzing superpotentials with weight vectors outside the convex hull, revealing new geometric structures and conditions.

## Key findings

- Isotropy representation has at most 3 irreducible summands in new cases
- First order subsystem resembles Calabi-Yau conditions
- Identifies geometric structures related to complex line bundles over Fano Kähler-Einstein products

## Abstract

We extend our previous classification of superpotentials of ``scalar curvature type" for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in our previous paper, i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano K\"ahler-Einstein product.