Understanding the Kuranishi map π for the heterotic (3) moduli space

Investigate and characterize the obstruction (Kuranishi) map π: U → O introduced in Theorem 1.1 for the local moduli space of solutions to the heterotic (3) system near a fixed solution s0, determining its properties that govern whether the moduli space is zero- or positive-dimensional in practice.

Background

The main theorem shows that infinitesimal deformations and obstructions have equal dimensions, implying an expected zero-dimensional moduli space, with local structure encoded by a map π between equal-dimensional spaces.

However, the actual geometry depends critically on π; if π vanishes or has non-generic behavior, positive-dimensional moduli may occur, so understanding π is central to determining the true local structure.