# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  [ABS] Andreatta, M., Ballico, E., Sommese, A.: On the projective normality of adjunction bundles, II (to appear)  [AS] Andreatta, M., Sommese, A.: On the projective normality of the adjunction bundles I (to appear) · Zbl 0758.14035  [BEL] Bertram, A., Ein, L., Lazarsfeld, R.: Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. Am. Math. Soc.4, 587-602 (1991) · Zbl 0762.14012 · doi:10.1090/S0894-0347-1991-1092845-5  [B] Butler, D.: Normal generation of vector bundles over a curve. J. Differ. Geom. (to appear) · Zbl 0808.14024  [CGGH] carlson, J., Green, M., Griffiths, P., Harris, J.: Infinitesimal variations of Hodge structure, I. Compos. Math.50, 109-205 (1983) · Zbl 0531.14006  [D] Demailly, J.-P.: A numerical criterion for very ample line bundles (to appear) · Zbl 0783.32013  [E] Ein, L.: The ramification divisors for branched coverings ofP n , Math. Ann.261, 483-485 (1982) · Zbl 0519.14005 · doi:10.1007/BF01457451  [EG] Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Albebra88, 89-133 (1984) · Zbl 0531.13015 · doi:10.1016/0021-8693(84)90092-9  [G1, G2] Green, M.: Koszul cohomology and the geometry of projective varieties, I, II. J. Differ. Geom.19, 125-171 (1984);20, 279-289 (1984)  [G3] Green, M.: The period map for hypersurface sections of high degree of an arbitrary variety. Compos. Math.55, 135-156 (1984)  [G4] Green, M.: A new proof of the explicit Noether Lefschetz theorem. J. Differ. Geom.27, 155-159 (1988) · Zbl 0674.14005  [G5] Green, M.: Koszul cohomology and Geometry. In: Cornalba, M. et al. (eds.) Lectures on Riemann Surfaces, Singapore: World Scientific Press 1989 · Zbl 0800.14004  [GL1] Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math.83, 73-90 (1986) · Zbl 0594.14010 · doi:10.1007/BF01388754  [GL2] Green, M., Lazarsfeld, R.: Some results on the syzygies of finite sets and algebraic curves. Compos. Math.67, 301-314 (1988) · Zbl 0671.14010  [GLP] Gruson, L., Lazarsfeld, R., Peskine, C.: On a theorem of Castelnuovo and the equations defining projective varieties. Invent. Math.72, 491-506 (1983) · Zbl 0565.14014 · doi:10.1007/BF01398398  [EGA] Grothendieck, A., Dieudonné, J.: Elements de géometrie algébrique, III. Publ. Math., Inst. Hautes Etud. Sci.17 (1963)  [K] Kempf, G.: The projective coordinate ring of abelian varieties. In: Igusa, J.I. (ed.) Algebraic Analysis, Geometry and Number Theory, pp. 225-236. Baltimore: Johns Hopkins Press 1989  [Kim] Kim, S.-O.: Noether-Lefschetz locus for surfaces. Trans. Am. Math. Soc.324, 369-384 (1991) · Zbl 0739.14019 · doi:10.2307/2001513  [L1] Lazarsfeld, R.: A sharp Castelnuovo bound for smooth surfaces. Duke Math. J.55, 423-429 (1987) · Zbl 0646.14005 · doi:10.1215/S0012-7094-87-05523-2  [L2] Lazarsfeld, R.: A sampling of vector bundle techniques in the study of linear series. In: Cornalba, M. et al. (eds) Lectures on Riemann Surfaces, pp. 500-559. Singapore: World Scientific Press 1989 · Zbl 0800.14003  [LeP] LePotier, J.: Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque. Math. Ann.218, 35-53 (1975) · Zbl 0313.32037 · doi:10.1007/BF01350066  [Man] Manivel, L.: Un théorem d’annulation pour les puissances extérieures d’un fibré ample (to appear)  [Mois] Moisezon, B.: Algebraic homology classes on algebraic varieties. Izv. Akad. Nauk. SSSR31, 225-268 (1976)  [M1] Mumford, D.: Lectures on curves on an algebraic surface. (Ann Math. Stud., no. 59) Princeton: Princeton University Press & University of Tokyo Press 1966  [M2] Mumford, D.: Varieties defined by quadratic equations. Corso CIME 1969. In: Questions on algebraic varieties, pp. 30-100. Rome: Eglizione cremonese 1970  [P] Pareschi, G.: Koszul algebras associated to adjunction bundles. J. Algebra (to appear) · Zbl 0796.13012  [R] Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math.127, 309-316 (1988) · Zbl 0663.14010 · doi:10.2307/2007055  [Sch] Schneider, M.: Ein einfacher Beweis des Verschindungssatzes für polsitive holomorphe Vectorraumbundel. Manuscr. Math.11, 95-101 (1974) · Zbl 0275.32014 · doi:10.1007/BF01189093  [SS] Shiffman, B., Sommese, A.: Vanishing Theorems on Complex Manifolds. (Prog. Math., vol. 56) Boston Basel Stuttgfart: Birkhäuser 1985 · Zbl 0578.32055  [S] Sommese, A.: Submanifolds of abelian varieties. Math. Ann.233, 229-256 (1978) · Zbl 0381.14007 · doi:10.1007/BF01405353  [Sed] Sommese, A. et al (eds).: Algebraic Geometry. Proceedings, L’Aquila 1988. (Lect. Notes Math., vol. 1417) Berlin Heidelberg New York: Springer 1990
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.