UDC 513.013.7

MATHEMATICS

G. N. TYURINA

ON ONE TYPE OF CONTRACTIBLE CURVES

(Presented by Academician L. S. Pontryagin, 12 V 1966)

Let the curve \(A=\cup A_i\), where the \(A_i\) are irreducible components, lie on a nonsingular complex surface \(X\). The curve \(A\) is called contractible (exceptional) if there exists a holomorphic mapping \(\pi\) of the surface \(X\) into a complex space \(Y\), which takes the whole curve \(A\) to one point \(p\in Y\) and is biholomorphic on \(X\setminus A\). In Grauert’s paper \((^1)\) it was proved that the curve \(A\) is contractible if and only if the intersection matrix \((A_i,A_j)\) is negative definite. In the present note we consider contractible curves preserving the canonical class of the surface, i.e. such that the canonical bundle of the surface \(X\) is trivial in some neighborhood of the curve \(A\). M. Artin proved in \((^2)\) that for this it is necessary and sufficient that each component \(A_i\) of the curve \(A\) be a nonsingular rational curve with \(A_i^2=-2\), and that in this case the curve \(A\) is contractible also in the algebraic sense. As was shown in another paper of M. Artin \((^3)\), the result of contracting such curves is rational double singularities, and only these.

In the present note it is shown that rational double points are nonvarying, and if neighborhoods of two rational double points are homeomorphic, then they are also biholomorphically equivalent. It follows that any rational double point can be realized as a singular point on a surface \(V\) in three-dimensional space, defined by an equation \(f(x,y,z)=0\) of a certain form. For the resolution of this singularity it proves sufficient to apply a sequence of \(\sigma\)-processes with centers at singular points of multiplicity 2. Let \(\widetilde V\) be the nonsingular surface obtained after such a resolution. From the last theorem of the present note it follows that the surface \(\widetilde V\) is diffeomorphic to the nonsingular surface \(f(x,y,z)=c\).

1. Let the curve \(A=\cup A_i\), corresponding to a rational double point, lie on a nonsingular surface \(X\), and let the curve \(\widetilde A=\cup \widetilde A_i\) isomorphic to it lie on a nonsingular surface \(\widetilde X\).

Theorem 1. Neighborhoods of the curve \(A\) on the surface \(X\) and of the curve \(\widetilde A\) on the surface \(\widetilde X\) are biholomorphically equivalent.

Proof. Let \(\mathfrak m_i\) be the sheaf of ideals on the surface \(X\) corresponding to the curve \(A_i\), and \(\mathcal O(X)\) the sheaf of germs of holomorphic functions on \(X\).

Denote by \((A,\mathcal O_Z)\) the complex space with structure sheaf
\[ \mathcal O_Z=\mathcal O_X/\mathfrak m(Z)\mid A, \]
where \(Z=\sum k_i A_i,\ k_i>0\), and
\[ \mathfrak m(Z)=\mathfrak m_1^{k_1}\mathfrak m_2^{k_2}\ldots \mathfrak m_n^{k_n}. \]

As follows from Theorem 3 of \((^4)\), in order to prove Theorem 1 it is enough to show that the complex spaces \((A,\mathcal O_{NA})\) and \((\widetilde A,\mathcal O_{N\widetilde A})\) are isomorphic for sufficiently large \(N\). The isomorphism of the curves \(A\) and \(\widetilde A\) is an isomorphism of the complex spaces \((A,\mathcal O_A)\) and \((\widetilde A,\mathcal O_{\widetilde A})\).

Suppose that some isomorphism
\[ \varphi:\ (A,\mathcal O_Z)\to(\widetilde A,\mathcal O_{\widetilde Z}) \]
has been established. How can one establish an isomorphism of the complex spaces \((A,\mathcal O_{Z+A_i})\) and \((\widetilde A,\mathcal O_{\widetilde Z+\widetilde A_i})\)? The obstruction \(\gamma(\varphi)\) to extending the given isomorphism \(\varphi\), as computed in \((^1)\), lies in the group
\[ H^1(A_i,\Theta\otimes(-Z,A_i)), \]
where \(\Theta\) is the restric-

of the tangent bundle of the surface \(X\) on the curve \(A_i\), and \(-Z \cdot A_i\) is the bundle corresponding to the cycle \(-Z\), restricted to \(A_i\). If this group is trivial, then there exists an isomorphism of complex spaces \((A,\mathcal O_{Z+A_i})\) and \((\widetilde A,\mathcal O_{\widetilde Z+\widetilde A_i})\) extending the given isomorphism \(\varphi\). Suppose that this group is nontrivial and \(\gamma(\varphi)\ne 0\). Consider the mapping

\[ \delta_Z:\operatorname{Aut}(A,\mathcal O_Z)\to H^1(A_i,\Theta\otimes(-Z,A_i)), \]

which assigns to each automorphism of the complex space \((A,\mathcal O_Z)\) the obstruction to extending it to an automorphism of the space \((A,\mathcal O_{Z+A})\). If \(\delta_Z\alpha=\gamma(\varphi)\), then the isomorphism \(\varphi\alpha^{-1}\) can be extended. If the image of the mapping \(\delta_Z\) coincides with the group \(H^1(A_i,\Theta\otimes(-Z,A_i))\), then from the isomorphism of the complex spaces \((A,\mathcal O_Z)\) and \((\widetilde A,\mathcal O_{\widetilde Z})\) there follows an isomorphism of \((\widetilde A,\mathcal O_{Z+A_i})\) and \((\widetilde A,\mathcal O_{\widetilde Z+\widetilde A_i})\).

In the following two lemmas we consider contractible curves corresponding to rational singular points (for the definition see (2)). By \(K\) is denoted the canonical divisor on the surface \(X\). In the case of a double rational point, \(K\cdot A_i=0\) for every \(i\). It is also assumed that the intersection matrices of the curves \(A\) and \(\widetilde A\) coincide.

Lemma 1. If \((Z+K)A_i < A_i(A-A_i)\) and there exist curves \(A_k\), where \(k=1,2,\ldots,(Z+K)A_i+1\), such that \(A_k\cdot A_i=1\) and \((Z+K)\cdot A_k\le 0\), then the image of the mapping \(\delta_Z\) coincides with the group \(H^1(A_i,\Theta\otimes(-Z,A_i))\).

Lemma 2. If \((Z+K)A_i\le 1\) and there exist curves \(A_1,\ldots,A_k\) such that \(A_i\cdot A_1=A_1\cdot A_2=\cdots=A_{k-1}\cdot A_k=1\) and \(((Z+K)-B)\cdot A_l\le 0\), where \(B=A_1+A_2+\cdots+A_k\), for all \(l=1,\ldots,k;\ i\), then the image \(\delta_Z\) coincides with the whole group \(H^1(A_i,\Theta\otimes(-Z,A_i))\).

Remark 1. If \(k=0\), then the lemma asserts the surjectivity of \(\delta_Z\) in the case \((Z+K)A_i=0\).

Remark 2. In considering analogous questions for algebraic varieties over a field of characteristic \(p\ne 0\), Lemma 1 remains valid for any \(p\), while Lemma 2 is valid only for \(p\) relatively prime to the determinant \(|A_l\cdot A_j|\), where \(l,j=1,\ldots,k;\ i\).

For curves corresponding to double rational points, one can construct a chain of cycles \(Z_{m+1}=Z_m+A_{i(m)}\), satisfying, for every \(m\), either Lemma 1 or Lemma 2, beginning with the cycle \(Z_0=A\), and such that for every \(N\) there is a cycle \(Z_m\) in the chain satisfying the inequality \(Z_m>NA\). Hence the assertion of Theorem 1 follows. By computing the fundamental group it is not hard to show that the boundaries of neighborhoods of different rational double points are not homeomorphic.

2. In (2) all possible configurations of the curves under consideration are listed. It can be shown that, resolving the singularity given in \(C^3\) by the equation \(x^2+y^2+z^n=0\), we obtain a curve of type \(A_{n-1}\); the equation \(x^2+y^2z+z^n=0\) corresponds to the curve \(B_{n+1}\); the equations \(x^2+y^3+z^4=0\), \(x^2+y^3+yz^3=0\), \(x^2+y^3+z^5=0\) correspond to the curves \(E_6\), \(E_7\), and \(E_8\). From Theorem 1 it then follows that a sufficiently small neighborhood of any rational double point can be biholomorphically embedded in \(C^3\) and is defined there by one of the equations given above. It is also easy to observe that, for resolving these singularities, it is enough to apply \(\sigma\)-processes with centers at singular points of multiplicity 2. It is not hard to show the converse as well: if, in resolving some singularity on a surface in \(C^3\), only \(\sigma\)-processes with centers at singular points of multiplicity 2 are applied, then it is a rational double point.

3. Consider in the space \(C^3\) the surface \(F_c\) defined by the equation \(f(x,y,z)=c\). Suppose that for \(c\ne 0\) the surface \(F_c\) is nonsingular, while the surface \(F_0\) has a unique singular point at the origin. Let \(\widetilde F_0\) be the nonsingular surface obtained as the result of resolving the singularity on \(F_0\).

Theorem 2. If, in resolving the singularity on the surface \(F_0\), only \(\sigma\)-processes with centers at singular points of multiplicity 2 are applied, then the surface \(\widetilde F_0\) is diffeomorphic to the surface \(F_c\).

Proof. In the simplest case, when \(f(x,y,z)=x^2+y^2+z^2\), the required diffeomorphism is easily constructed explicitly. In the general case we carry out induction on the number of \(\sigma\)-processes needed to resolve the singularity. Suppose, for definiteness, that the quadratic part of the function \(f(x,y,z)\) has rank 1; then we may assume that
\[ f(x,y,z)=x^2+g(x,y,z), \]
where the function \(g(x,y,z)\) has a zero of multiplicity 3 at the origin. Consider the family of surfaces \(F(c_1,c_2,c)\) given by the equations
\[ f(x,y,z)=c_1y^2+c_2z^2+c. \]
The nonsingular surfaces \(F(c_1,c_2,c)\) are diffeomorphic to one another and, in particular, to the surfaces \(F_c=F(0,0,c)\). For \(c_1=c_2=c=0\) we obtain the surface \(F_0\); for \(c=0,\ c_1\ne0,\ c_2\ne0\) we obtain a surface with one quadratic singularity. Apply a \(\sigma\)-process with center at the origin. The surface \(\sigma F_0\) has singularities whose resolution requires applying a smaller number of \(\sigma\)-processes. Using the induction hypothesis, one can show that the singular surface \(\tilde F_0\) is diffeomorphic to the nonsingular surface \(\sigma F(c_1,c_2,0)\overset{\mathrm{diff}}{\approx}F(c_1,c_2,c)\) for \(c_1\) and \(c_2=0\); hence the assertion of the theorem follows.

1. Let \(\mathcal V\) be a singular \(n\)-dimensional complex variety, decomposing by means of a holomorphic mapping \(\pi:\mathcal V\to T\) into complex surfaces \(V_t\). Suppose that the base \(T\) is nonsingular, and all fibers \(V_t\) are either nonsingular or have only double rational singularities, while the set \(S\subset T\) of those points \(t\) for which the surface \(V_t\) is singular is a complex subspace of smaller dimension. Let \(s\in S\). Resolve the singularities on \(V_s\) minimally. We obtain the surface \(\tilde V_s\). From Theorems 1 and 2 one can derive the following assertion:

Theorem 3. The surface \(\tilde V_s\) is diffeomorphic to the nonsingular surface \(V_t\).

In conclusion, the author expresses his gratitude to I. R. Shafarevich and Yu. I. Manin for their attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
27 V 1966

REFERENCES

1. H. Grauert, Math. Ann., 146, 331 (1962); Collection, Complex Spaces, Mir, 1965.
2. M. Artin, Am. J. Math., 84, No. 3, 485 (1962); Collection of translations, Mathematics, 9, 3 (1965).
3. M. Artin, Am. J. Math., 88, No. 1, 129 (1966).
4. H. Hironaka, H. Rossi, Math. Ann., 156, 313 (1964).