Several complex variables and analytic spaces Complex singularities Theorem

Mather–Yau theorem

Builds on Isolated hypersurface singularity Ring of convergent power series Ideal

Statement

Let $\mathcal{O}_{n+1} = \C\{z_0, \dots, z_n\}$ be the ring of convergent power series. For $f \in \mathcal{O}_{n+1}$ with $f(0) = 0$ , let the Tjurina algebra be $A(f) = \mathcal{O}_{n+1} \Big/ \left( f, \frac{\partial f}{\partial z_0}, \dots, \frac{\partial f}{\partial z_n} \right),$ the quotient by the ideal generated by $f$ and its partial derivatives. Let $f, g \in \mathcal{O}_{n+1}$ have isolated singularities at $0$ . Then the following are equivalent:

the hypersurface germs $(f^{-1}(0), 0)$ and $(g^{-1}(0), 0)$ are biholomorphic, that is, there are a biholomorphic germ $\phi$ of $(\C^{n+1}, 0)$ and a unit $u \in \mathcal{O}_{n+1}$ with $g = u \cdot (f \circ \phi)$ ;
$A(f)$ and $A(g)$ are isomorphic as $\C$ -algebras.

Remarks

John Mather and Stephen Yau proved the theorem in 1982. The direction from 1 to 2 is a direct computation, since the ideal $(f, \partial f / \partial z_j)$ is carried to $(g, \partial g / \partial z_j)$ by $\phi$ and $u$ . The converse is the hard part. Its proof uses the finite determinacy of isolated singularities and a study of the action of the group of coordinate changes on jets.

Example: for $f = x^2 + y^{k+1}$ with $k \ge 1$ , the ideal is $(x, y^k)$ , so $A(f) \cong \C[y] / (y^k)$ . These algebras have different dimensions for different $k$ , so the singularities $A_k$ are pairwise not biholomorphic.

The dimension $\tau(f) = \dim_\C A(f)$ is the Tjurina number. It satisfies $\tau(f) \le \mu(f)$ , and K. Saito proved that $\tau(f) = \mu(f)$ if and only if $f$ is right equivalent to a weighted homogeneous polynomial.

Going deeper

For a student

The Tjurina algebra is also the space of first-order deformations of the hypersurface germ: its dimension $\tau$ is the number of parameters of the miniversal deformation. The theorem says that this algebra remembers the whole singularity. Together with the Milnor number and the classification of simple singularities, it is a basic tool of the analytic classification of isolated hypersurface singularities.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Ideal concept
Isolated hypersurface singularity concept
Ring of convergent power series concept
Mather–Yau theorem statement

Tag	`QMMSKHEW9S`
URI	`m2://QMMSKHEW9S`
Kind	statement, theorem
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MSC	32S05, 32S25, 14B05
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About	Isolated hypersurface singularity
Uses	Isolated hypersurface singularity, Ring of convergent power series, Ideal