Let \[ X\subseteq\AA^n_\CC \] be an affine variety of complex dimension \(d\), and let \(p\in X\). Show that \(p\) is smooth if and only if, in a neighborhood of \(p\), \[ X\subseteq\CC^n\cong\RR^{2n} \] is a smooth real submanifold of dimension \(2d\).
Solution
Choose generators \[ I(X)=(f_1,\ldots,f_m) \] and write \[ F=(f_1,\ldots,f_m):\CC^n\longrightarrow\CC^m. \] Let \[ J_\CC(p) = \left( \frac{\partial f_i}{\partial x_j}(p) \right) \] be the complex Jacobian matrix.
1. The point \(p\) is algebraically smooth exactly when\[ \rank_\CC J_\CC(p)=n-d. \]
This is the Jacobian criterion PR-MORJAC for the affine variety \(X\).
2. Viewed as a real map\[ F_\RR:\RR^{2n}\longrightarrow\RR^{2m}, \]the differential at \(p\) satisfies\[ \rank_\RR dF_\RR(p) = 2\rank_\CC J_\CC(p). \]
The differential \[ dF(p):\CC^n\longrightarrow\CC^m \] is complex linear because each \(f_i\) is holomorphic. A complex-linear map of complex rank \(r\) has image a complex vector space of complex dimension \(r\), hence a real vector space of dimension \(2r\). Therefore its underlying real linear map has rank \(2r\).
3. If \(p\) is algebraically smooth, then \(X\) is a smooth real submanifoldof dimension \(2d\) near \(p\).
Assume \(p\) is smooth. By step 1, \[ \rank_\CC J_\CC(p)=n-d. \] Choose \(n-d\) of the defining functions whose differentials are complex linearly independent at \(p\), and write \[ G:\CC^n\longrightarrow\CC^{n-d} \] for the resulting submap.
By step 2, \[ \rank_\RR dG_\RR(p)=2(n-d). \] The real Implicit Function Theorem therefore makes \[ G^{-1}(0) \] a smooth real submanifold of dimension \[ 2n-2(n-d)=2d \] near \(p\).
The holomorphic Implicit Function Theorem makes \(G^{-1}(0)\), near \(p\), a connected complex submanifold \(M\) of dimension \(d\). It contains \(X\) near \(p\), since the components of \(G\) are among the \(f_i\). A closed analytic subset of dimension \(d\) of a connected \(d\)-dimensional complex manifold is the whole manifold, so \(X=M\) near \(p\), and \(X\) is a smooth real submanifold of dimension \(2d\) there.
4. If \(X\) is a smooth real submanifold \(M\) of dimension \(2d\) near \(p\),then \(X\) is a complex submanifold of dimension \(d\) near \(p\).
The smooth points of \(X\) are dense in \(X\). At a smooth point \(q\) near \(p\), step 3 makes \(X\) a complex submanifold of dimension \(d\) equal to \(M\) near \(q\), so \(T_qM\) is a complex subspace of \(\CC^n\). Complex subspaces of real dimension \(2d\) form a closed set, and \(T_qM\) depends continuously on \(q\), so \(T_pM\) is a complex subspace too.
Let \(\pi\) be the complex-linear projection of \(\CC^n\) onto \(T_pM\) along a complex complement \(N\). Then \(d\pi\) is the identity on \(T_pM\), so near \(p\), \(M\) is the graph of a smooth map \(\varphi\) from an open set of \(T_pM\cong\CC^d\) to \(N\). At a smooth point \(q\) of \(X\) near \(p\), \(M\) is a complex submanifold whose tangent space projects isomorphically to \(T_pM\), so \(\varphi\) is holomorphic near \(\pi(q)\) by the holomorphic Inverse Function Theorem. So the derivative of \(\varphi\) is complex linear on a dense set, hence everywhere by continuity, and \(\varphi\) is holomorphic. Its graph \(M\) is a complex submanifold of dimension \(d\).
5. Under the hypothesis of step 4, the point \(p\) is algebraicallysmooth.
By step 4 the analytic local ring of \(X\) at \(p\) is the ring of convergent power series in \(d\) variables. The algebraic local ring \(\OO_{X,p}\) and the analytic local ring have isomorphic completions [@Mum94, §I.10], so \(\widehat{\OO_{X,p}}\cong\CC[[z_1,\ldots,z_d]]\) is regular. A noetherian local ring is regular exactly when its completion is, so \(\OO_{X,p}\) is regular of dimension \(d\), and \(p\) is smooth.
6. Therefore\[ \boxed{ p\text{ is smooth} \quad\Longleftrightarrow\quad X\text{ is locally a smooth real }2d\text{-manifold near }p. } \]
Step 3 proves the forward implication, and steps <1>4–<1>5 prove the converse.
7. Q.E.D.
Step 6 is the required equivalence.