B. G. MOISHEZON

ON PROJECTIVE EMBEDDINGS OF ALGEBRAIC VARIETIES

(Presented by Academician P. S. Novikov, 16 VI 1961)

MATHEMATICS

The well-known theorem of Kodaira \((^1)\) asserts that if a divisor \(D\) on a compact Kähler manifold \(V\) is such that in the class of differential forms corresponding to the cohomology class dual to it there is found a form of type \((1,1)\),

\[ \sum h_{\alpha\beta}\, dz_\alpha \wedge d\overline{z}_\beta \]

such that the matrix \(\|h_{\alpha\beta}\|\) is positive definite at every point of the manifold \(V\), then the mapping of the manifold \(V\) into projective space corresponding to a sufficiently high multiple of the divisor \(D\) is a biregular embedding, i.e., it realizes the manifold \(V\) as a certain projective algebraic variety.

The question arises: what corresponds to this theorem in algebraic geometry?

Nakai \((^2)\) showed that, in order that a sufficiently high multiple of a divisor \(D\) on a nonsingular algebraic surface \(V_2\) correspond to a biregular embedding of the surface \(V_2\) in projective space, it is necessary and sufficient that the following conditions be satisfied:

\[ D_2 > 0,\qquad D \cdot C > 0, \]

where \(C\) runs through all one-dimensional subvarieties of the surface \(V_2\).

We shall assume that the ground field \(k\) is algebraically closed of characteristic zero. Let \(U_r\) be a nonsingular complete abstract algebraic variety of dimension \(r\).

Theorem 1. Let \(D\) be a divisor on \(U_r\) such that \(D^r > 0\) and \(D^i \cdot C_i > 0\), where \(i = 1, \ldots, r - 1\), and \(C_i\) runs through all subvarieties of the variety \(U_r\) having dimension \(i\). Then, for sufficiently large \(n\), the mapping of the variety \(U_r\) into projective space corresponding to the complete linear system \(|nD|\) is a biregular embedding.

The following corollaries follow from Theorem 1:

Corollary 1. In order that a nonsingular complete abstract algebraic variety \(U_r\) be projective, it is necessary and sufficient that there exist on \(U_r\) a divisor \(D\) such that

\[ D^r > 0,\qquad D^i \cdot C_i > 0 \quad (i = 1,\ldots,r-1) \]

for all \(i\)-dimensional subvarieties \(C_i\) of the variety \(U_r\).

Corollary 2. Let \(D\) be a divisor on a nonsingular algebraic variety \(U_r(\mathbb{C})\) over the field of complex numbers \(\mathbb{C}\). Then, in order that in the class of differential forms corresponding to the dual—

to the divisor \(D\) in the cohomology class, there is found a form of type \((1,1)\)

\[ \sum h_{\alpha\beta}dz_{\alpha}\wedge d\bar z_{\beta} \]

such that the matrix \(\|h_{\alpha\beta}\|\) is positive definite at each point of the variety \(U_r(C)\), it is necessary and sufficient that the conditions

\[ D^r>0,\qquad D^i\cdot C_i>0\quad (i=1,\ldots,r-1) \]

hold for all \(i\)-dimensional subvarieties \(C_i\) of the variety \(U_r(C)\).

Induction on the dimension is not directly applicable to the proof of Theorem 1. Therefore one has to prove a more general theorem, from which Theorem 1 is obtained as a special case:

Theorem 2. Let \(V_s\) be a complete abstract (possibly reducible) algebraic variety.

Suppose that there exists a regular mapping \(f:V_s\to U_r\) such that to each point of the image \(fV_s\) there corresponds only a finite number of points on \(V_s\). Suppose that a divisor \(C\) on \(U_r\) is such that \(D^s\cdot fV_s>0\), \(D^i\cdot fC_i>0\) \((i=1,\ldots,s-1)\) for all \(i\)-dimensional subvarieties \(C_i\) of the variety \(V_s\) (where \(fC_i\) denotes the image of \(C_i\) under the mapping \(f\)). To the divisor \(nD\) there corresponds the fibred space of lines \(\{nD\}\) over \(U_r\), inducing a certain fibred space over \(V_s\). Denote the latter by \(\{nD\}_{V_s}\).

Then, for sufficiently large \(n\), the mapping of the variety \(V_s\) into projective space, given by the group of sections of the fibred space \(\{nD\}_{V_s}\), is a biregular embedding.

We give a brief outline of the proof of Theorem 2. The passage from \(s-1\) to \(s\) is carried out in the following way. First the assertion of Theorem 2 is verified for normal \(V_s\), then for irreducible ones, and after that for reducible ones.

An important role in the passage from normal varieties to irreducible ones is played by the following consideration. If \(V_s^*\) is the normalization of the irreducible variety \(V_s\), then on \(V_s\) there exists such a positive divisor \(E\) that, if by \(E^*\) we denote its inverse image on \(V_s^*\), by

\[ \overline{O_{V_s^*}(-E^*)} \]

the sheaf over \(V_s^*\) whose stalk at a point \(P\in V_s^*\) consists of functions \(F\) such that \((F)-E^*>0\) locally at the point \(P\), and by

\[ \overline{O_{V_s}^*(-E^*)} \]

the direct image of the sheaf \(O_{V_s^*}(-E^*)\) corresponding to the mapping \(V_s^*\to V_s\), then there is an exact sequence

\[ 0\to \overline{O_{V_s^*}(-E^*)}\to O_{V_s}\to K\to 0, \]

where \(O_{V_s}\) is the sheaf of local rings of the variety \(V_s\), and \(K\) is a certain coherent sheaf over \(V_s\), concentrated on the support of the divisor \(E\). With the help of an analogous consideration one makes the passage from irreducible varieties to reducible ones.

After it has been proved that the mapping \(f_n\), given by the group of sections of the fibred space \(\{nD\}_{V_s}\), is regular and maps distinct points of \(V_s\) to distinct points of projective space, we argue as follows. Let \(V_s^{(n)}\) be the image of \(V_s\) under the mapping \(f_n\). \(V_s^{(n)}\) is a projective algebraic variety. Obviously, \(f_n\) realizes a homeomorphism (in the Zariski topology) between \(V_s\) and \(V_s^{(n)}\). Denote by \(\bar O_{V_s}\) the image of the sheaf \(O_{V_s}\) over \(V_s^{(n)}\). The sheaf obtained is coherent. Denote by \(\bar O_{V_s}(m)\) the sheaf obtained from \(\bar O_{V_s}\) by \(m\)-fold application of Serre’s operation \(F(m)\) (see (3)). It is not hard to see that the sheaf corresponding to the sheaf \(\bar O_{V_s}(m)\) over \(V_s\) coincides with the sheaf

local sections of the fibered space \(\{mnD\}_{V_s}\). Let \(\alpha \in O_{V_s}(P)\) (\(O_{V_s}(P)\) is the local ring of the point \(P \in V_s\)); \(y\) is a section of the fibered space \(\{mnD\}_{V_s}\), not equal to zero at \(P\); \(y_1,\ldots,y_N\) is a basis of the group of sections of the fibered space \(\{mnD\}_{V_s}\), and let \(\bar\alpha,\bar y,\bar y_1,\ldots,\bar y_N\) be the corresponding elements over \(V_s^{(n)}\). By the well-known theorem of Serre \((^3)\), for sufficiently large \(m\) the coherent sheaf \(F(m)\) over the projective algebraic variety \(V\) is generated by the elements of the group \(H^0(V,F(m))\). Hence, if \(m\) is sufficiently large, then \(\bar\alpha \bar y=\sum \beta_i \bar y_i\), where \(\beta_i\) belong to the local ring \(O_{V_s^{(n)}}(f_nP)\) (\(f_nP\) is the image of the point \(P\) on \(V_s^{(n)}\)). Hence

\[ \bar\alpha=\sum \beta_i \frac{\bar y_i}{\bar y}. \]

Consider the mapping \(f_{mn}\). There exists a regular mapping \(t_{m,n}\) such that \(t_{m,n}f_{mn}=f_n\):

\[ \begin{array}{ccc} & V_s^{(mn)} & \\ {}^{f_{mn}}\swarrow & & \searrow^{t_{m,n}} \\ V_s & \xrightarrow{\ f_n\ } & V_s^{(n)} \end{array} \]

and, consequently, \(\beta_i\) belong to the local ring \(O_{V_s^{(mn)}}(f_{mn}P)\).

Obviously, \(\dfrac{\bar y_1}{\bar y},\ldots,\dfrac{\bar y_N}{\bar y}\) may be regarded as nonhomogeneous coordinates in a certain affine neighborhood of the point \(f_{mn}P\). It follows that \(\alpha \in O_{V_s^{(mn)}}(f_{mn}P)\), i.e. the sheaves \(\bar O_{V_s}\) and \(O_{V_s^{(mn)}}\) coincide. This proves that the mapping \(f_{mn}\) is biregular.

I express my deep gratitude to my adviser I. I. Pyatetskii-Shapiro.

References Cited

\(^1\) K. Kodaira, Ann. of Math., 60, 28 (1954).
\(^2\) Y. Nakai, J. Sci. Hirosima Univ., Ser. A, 24, No. 1, Juli (1960).
\(^3\) J.-P. Serre, Ann. Math., 61, 197 (1955).