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Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. (English) Zbl 0814.14040
The equations defining a projective variety \(X \subset \mathbb{P}^ r\) and the syzygies among them have attracted considerable attention. Let \(X\) be a smooth complex projective variety of dimension \(n\), and let \(L\) be a very ample line bundle on \(X\), defining an embedding \(X \subset \mathbb{P} = \mathbb{P} H^ 0(L)\). Denote by \(S = \text{Sym}^ \bullet H^ 0(L)\) the homogeneous coordinate ring of the projective space \(\mathbb{P}\), and consider the graded \(S\)-module \(R = R(L) = \bigoplus H^ 0(X,L^ d)\). Let \(E_ \bullet\) be a minimal graded free resolution of \(R\): \[ \begin{matrix}\ldots & \to \bigoplus S(-a_{2,j}) & \to \bigoplus S(-a_{1,j}) & \to \bigoplus S(-a_{0,j}) & \to R \to 0.\\ & \| & \| & \| \\ & E_ 2 & E_ 1 & E_ 0 \end{matrix} \] \(R\) has a canonical generator in degree zero. Observe that all \(a_{0,j} \geq 2\) since we are dealing with a linearly normal embedding, and that all \(a_{i,j} \geq i + 1\) when \(i \geq 1\) thanks to the fact that \(X\) does not lie on any hyperplanes. We shall be concerned with situations in which the first few modules of syzygies of \(R(L)\) are as simple as possible:
Definition. The line bundle \(L\) satisfies property \((N_ p)\) if \(E_ 0 = S\) when \(p \geq 0\) and \(E_ i = \bigoplus S(-i-1)\) (i.e. all \(a_{i,j} = i + 1\)) for \(1 \leq i \leq p\).
The definition may be summarized very concretely as follows:
\(L\) satisfies \((N_ 0) \Leftrightarrow X\) embeds in \(\mathbb{P} H^ 0(L)\) as a projectively normal variety;
\(L\) satisfies \((N_ 1) \Leftrightarrow (N_ 0)\) holds for \(L\), and the homogeneous ideal \(I\) of \(X\) is generated by quadrics;
\(L\) satisfies \((N_ 2) \Leftrightarrow (N_ 0)\) and \((N_ 1)\) hold for \(L\), and the module of syzygies among quadratic generators \(Q_ i \in I\) is spanned by relations of the form \(\sum L_ i \cdot Q_ i = 0\), where the \(L_ i\) are linear polynomials; and so on. Properties \((N_ 0)\) and \((N_ 1)\) are what Mumford calls respectively normal generation and normal presentation. – Our first main result is the following:
Theorem 1. Let \(X\) be a smooth complex projective variety of dimension \(n\), and let \(A\) be a very ample line bundle on \(X\). Then \((N_ p)\) holds for the bundle \(K_ X + (n + 1 + p)A\). More generally, if \(B\) is any numerically effective line bundle on \(X\), then \(K_ X + (n + 1 + p)A + B\) satisfies \((N_ p)\).
When \(p = 0\), the theorem asserts the projective normality of the embedding defined by \(K_ X + (n + 1)A + B\). It is well known that results on syzygies may be interpreted in terms of the vanishing of certain Koszul cohomology groups [cf. M. Green, J. Differ. Geom. 19, 125-171 and 20, 279-289 (1984; Zbl 0559.14008 and 14009)]. For \(X = \mathbb{P}^ n\), M. Green [loc. cit. and ibid. 27, No. 1, 155-159 (1988; Zbl 0674.14005)] proved a general vanishing theorem for such groups, which he, Voisin and others have used to make interesting infinitesimal computations in Hodge theory. In this spirit, we may view theorem 1 as a special case of the following:
Theorem 2. Let \(A\) be a very ample line bundle, and let \(B\) and \(C\) be numerically effective line bundles on a smooth complex projective \(n\)- fold \(X\). Put \(L_ d = K_ X + dA + B\) and \(N_ f = K_ X + fA + C\). Let \(W \subset H^ 0(X,L_ d)\) be a base-point free subspace of codimension \(c\), and consider the Koszul-type complex \[ \wedge^{p + 1} W \otimes H^ 0(N_ f) \to \wedge^ p W \otimes H^ 0(L_ d \otimes N_ f) \to \wedge^{p - 1} W \otimes H^ 0(L^ 2_ d \otimes N_ f). \] If \(d \geq n + 1\) and \(f \geq (n + 1) + p +c\), then this complex is exact (in the middle). Green’s result is the case \(X = \mathbb{P}^ n\) and \(A = {\mathcal O}_{\mathbb{P}^ n}(1)\).
We hope that theorem 2 may open the door to finding explicit formulations of theorems hitherto known precisely for \(\mathbb{P}^ n\) but only asymptotically in general. In this direction we prove in §3 the following: proposition 3.
Let \(X\) be a smooth complex projective threefold, and let \(A\) be a very ample and \(B\) a nef line bundle on \(X\).
(1) If \(Y \in | 3K_ X + 16A + B|\) is a sufficiently general smooth divisor, then \(\text{Pic}(Y) = \text{Pic}(X)\).
(2) If \(Y \in | K_ X + 8A + B|\) is any smooth divisor, then the infinitesimal Torelli theorem holds for \(Y\), i.e. the derivative of the period mapping is injective at \(Y\).
Note that when \(X = \mathbb{P}^ 3\), (1) is just the classical Noether- Lefschetz theorem and (2) is the elementary fact that infinitesimal Torelli holds for surfaces of degrees \(\geq 4\).

MSC:
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13D02 Syzygies, resolutions, complexes and commutative rings
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14A05 Relevant commutative algebra
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