The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1706–1743 | Cite as

Associated Forms: Current Progress and Open Problems

  • Alexander IsaevEmail author


Let \(d\ge 3\), \(n\ge 2\). The object of our study is the morphism \(\Phi \), introduced in earlier articles by J. Alper, M. Eastwood and the author, that assigns to every homogeneous form of degree d on \({\mathbb {C}}^n\) for which the discriminant \(\Delta \) does not vanish a form of degree \(n(d-2)\) on the dual space, called the associated form. This morphism is \({\mathrm{SL}}_n\)-equivariant and is of interest in connection with the well-known Mather–Yau theorem, specifically, with the problem of explicit reconstruction of an isolated hypersurface singularity from its Tjurina algebra. Letting p be the smallest integer such that the product \(\Delta ^p\Phi \) extends to the entire affine space of degree d forms, one observes that the extended map defines a contravariant. In the present paper, we survey known results on the morphism \(\Phi \), as well as the contravariant \(\Delta ^p\Phi \), and state several open problems. Our goal is to draw the attention of complex analysts and geometers to the concept of the associated form and the intriguing connection between complex singularity theory and invariant theory revealed through it.


Associated forms Isolated hypersurface singularities The Mather–Yau theorem Classical invariant theory Geometric Invariant Theory Contravariants of homogeneous forms 

Mathematics Subject Classification

13A50 14L24 32S25 



We are grateful to M. Fedorchuk for many very helpful discussions.


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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

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