The $E_6$ normal forms: semisimple deformations, geometry, DT invariants, and the $E_{6,\infty}$ order

Abstract

We study the completed Jacobi algebras of $x^3+xy^3+p(y)$ for $\operatorname{ord}_yp=n\ge4$. The explicit normal-form morsification, the filtered length and centre proofs, and the cyclic terminal identities give finite free semisimple deformations. For every parameter of finite order we construct a polynomial curve neighbourhood and a local projective small contraction; the resulting geometric GV and Donaldson--Thomas calculations identify the block multiplicities. The one-dimensional limit $E_{6,\infty}$ is studied separately as an order: we determine its normalization, conductor, radical-idealizer chain, and maximal overorders. The canonical chain yields a nine-dimensional finite quotient whose Ringel dual is isomorphic to its opposite. An expanded construction gives a choice-independent $22$-dimensional quotient and a $40$-dimensional Ringel dual; a corner quotient relates the two finite constructions. The length and centre formulas are proved by a filtered comparison with a homogeneous matrix order and by terminal identities at the exceptional endpoint.