MATHEMATICS

B. G. MOISHEZON

ON IRREDUCIBLE EXCEPTIONAL SUBMANIFOLDS OF THE FIRST KIND ON THREE-DIMENSIONAL COMPLEX-ANALYTIC MANIFOLDS

(Presented by Academician L. S. Pontryagin, October 3, 1964)

In the theory of algebraic surfaces, an exceptional curve of the first kind is a curve that can be contracted to a nonsingular point. The Castelnuovo–Enriques theorem, generalized to the analytic case by Grauert, asserts that for an irreducible curve \(S\) on a nonsingular surface \(F\) to be an exceptional curve of the first kind, it is necessary and sufficient that \(S\) be a nonsingular rational curve and \(S^{2}=-1\).

Definition. Let \(V\) be a complex-analytic manifold, \(\dim V=3\), and let \(D\) be an analytic subspace on \(V\). We shall call \(D\) an exceptional submanifold of the first kind on \(V\) if there exist a three-dimensional manifold \(V'\), a submanifold (nonsingular subspace) \(D'\) on it, and a regular proper mapping \(T:V\to V'\) such that \(T(D)=D'\), \(T\) establishes a biregular correspondence between \(V-D\) and \(V'-D'\), and the mapping \(T|_{D}:D\to D'\) is not biregular.

Theorem 1. Let \(V\) be a complex-analytic manifold, \(\dim V=3\), and let \(D\) be an irreducible subspace on \(V\) which is an exceptional submanifold of the first kind \((V',D',T\) as in the definition).

Then, if on \(V'\) one performs the \(\sigma\)-process along \(D'\) and denotes the embedded submanifold by \(\overline D\), the object obtained from \(V'\) by \(\overline V\), and the corresponding regular mapping \(\overline V\to V'\) by \(\overline T\), there exists a biregular correspondence between \(\overline V\) and \(V\), under which \(\overline D\) coincides with \(D\), and \(\overline T\) with \(T\).

Theorem 2. Let \(V\) be a complex-analytic manifold, \(\dim V=3\), and let \(D\) be a compact nonsingular ruled surface on \(V\), more precisely, a fibration into projective lines with base a compact nonsingular curve. Introduce the notation: \(l\) is a fiber of the fibration; \(C\) is the base; \(\pi:D\to C\) is the projection of the fibration onto the base. Let \(D\cdot l=-1\) (intersection index on \(V\)).

Then there exists a complex-analytic manifold \(V'\), on it a curve \(D'\), biregularly equivalent to the curve \(C\), and a regular mapping \(T:V\to V'\) such that \(T\) establishes a biregular correspondence between \(V-D\) and \(V'-D'\), and \(T|_{D}:D\to D'\) coincides with \(\pi:D\to C\).

Using Theorems 1 and 2, and also the Kodaira–Grauert condition \((^{1,2})\) for contractibility of a projective space to a nonsingular point, we obtain the following.

Criterion. For an irreducible compact subspace \(D\) on a three-dimensional complex-analytic manifold \(V\) to be an exceptional submanifold of the first kind, it is necessary and sufficient that \(D\) be either a projective plane with \(D\cdot e=-1\), where \(e\) is a line on \(D\), or a fibration into projective lines with nonsingular compact curve \(C\) as base and with \(D\cdot l=-1\), where \(l\) is a fiber of the fibration. In the first case the contraction coincides with the anti-\(\sigma\)-process to a nonsingular point, and in the second with the anti-\(\sigma\)-process onto a nonsingular curve. (By an anti-\(\sigma\)-process we here mean a mapping inverse to a \(\sigma\)-process.)

The question arises: if \(V\) is an algebraic variety, will the variety \(V'\), obtained after contracting the exceptional subvariety \(D\) mentioned in the criterion, always be algebraic?

For contraction to a point this follows from Kodaira’s result \({}^{1}\). As for contraction to a curve, it follows from Hironaka’s results \({}^{3}\) that \(V'\) may be neither algebraic nor even an abstract algebraic variety.

Indeed, on the one hand, examples are known of 3-dimensional complex-analytic varieties with three algebraically independent meromorphic functions which are not algebraic varieties (neither projective nor abstract); on the other hand, Hironaka proved that any such variety can, by a sequence of monoidal transformations with nonsingular centers, be made into a projective algebraic variety.

A rather direct example showing that, under contraction to a curve, a nonalgebraic variety can be obtained from an algebraic variety is constructed as follows.

In projective space \(P^{4}\) one must take a hypersurface \(F_{n}\) of degree \(n>2\) having one quadratic nondegenerate singular point \(O\) (all other points being nonsingular). Then the \(\sigma\)-process on \(P^{4}\) at the point \(O\) transforms \(F_{n}\) into a nonsingular variety \(\bar F_{n}\), and a quadric \((P^{1}\times P^{1})\) is glued into \(F_{n}\), which has intersection index \(-1\) with each of its line factors. If it is contracted onto either of these factors, then from \(\bar F_{n}\) one obtains a variety \(\bar{\bar F}_{n}\) which is neither projective, nor abstract algebraic, nor a Kähler variety.

REFERENCES CITED

\({}^{1}\) K. Kodaira, Ann. Math., 60, 28 (1954).
\({}^{2}\) H. Grauert, Math. Ann., 146, 331 (1962).
\({}^{3}\) H. Hironaka, Ann. Math., 79, No. 1 (1964).