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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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