# zbMATH — the first resource for mathematics

The infinitesimal Torelli problem for zero sets of sections of vector bundles. (English) Zbl 0613.14010
Let X be a compact complex manifold of dimension $$n$$ and consider an exact sequence $$0\to {\mathcal G}\to {\mathcal F}\to \Omega^ 1_ X\to 0.$$ Fix an integer p, $$0<p\leq n$$. The main result asserts that, assuming some cohomological conditions concerning the above data, the canonical map $$H^ 1(X,\Theta_ X)\to Hom(H^{n-p}(X,\Omega^ p_ X),H^{n- p+1}(X,\Omega_ X^{p-1}))$$ is injective. By Griffiths, it follows then that the infinitesimal Torelli theorem holds for X. Using this, the author obtains a general result concerning the infinitesimal Torelli problem for zero sets of sections of vector bundles (in rank one, this extends a result of Green). In particular from this one deduces that the infinitesimal Torelli theorem holds for complete intersections in projective spaces (with two precise exceptions), generalizing known results of Griffiths, Peters, Usui.
Reviewer: C.Bănică

##### MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14M10 Complete intersections 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
Full Text:
##### References:
 [1] Carlson, J., Green, M., Griffiths, Ph., Harris, J.: Infinitesimal variations of Hodge structures I. Comp. Math.50, 109-205 (1983) · Zbl 0531.14006 [2] Donagi, R.: Group law on the intersection of two quadrics. Ann. Scuola Norm. sup. Pisa Ser. IV, Vol. VII, 217-239 (1980) · Zbl 0457.14023 [3] Flenner, H.: Divisorenklassengruppen quasihomogener Singularit?ten. J. Reine Angew. Math.328, 128-160 (1981) · Zbl 0457.14001 · doi:10.1515/crll.1981.328.128 [4] Green, M.: The period map for hypersurface sections of high degree of an arbitrary variety. Preprint, University of California, Los Angeles · Zbl 0588.14004 [5] Griffiths, Ph.: Periods of integrals on algebraic manifolds I and II. Am. J. Math.90, 568-626 and 805-865 (1968) · Zbl 0169.52303 · doi:10.2307/2373545 [6] Griffiths, Ph.: Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems. Bull. AMS76, 228-296 (1970) · Zbl 0214.19802 · doi:10.1090/S0002-9904-1970-12444-2 [7] Griffiths, Ph., Harris, J.: Principles of algebraic geometry. Wiley, New York-Chichester-Brisbane-Toronto 1978 · Zbl 0408.14001 [8] Lebelt, K.: Freie Aufl?sungen ?u?erer Potenzen. Manuscripta Math21, 341-355 (1977) · Zbl 0365.13004 · doi:10.1007/BF01167853 [9] Liebermann, D., Peters, C., Wilsker, R.: A theorem of local Torelli type. Math. Ann.231, 39-45 (1977) · Zbl 0367.14006 · doi:10.1007/BF01360027 [10] Peters, C., Steenbrink, J.: Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces. In: Classification of algebraic and analytic manifolds, pp. 399-463. Proc. of the Katata Symp. 1982. Birkh?user, Boston-Basel-Stuttgart (1983) [11] Peters, C.: The local Torelli theorem I. Complete intersections. Math. Ann.217, 1-16 (1975); Erratum Math. Ann.223, 191-192 (1976) · Zbl 0304.14007 · doi:10.1007/BF01363236 [12] Peters, C.: The local Torelli theorem II. Cyclic branched coverings. Ann. Sc. Norm. Super. Cl. Sci., Pisa, Ser. IV.3, 321-340 (1976) · Zbl 0329.14011 [13] Schneider, M.: Some remarks on vanishing theorems for holomorphic vector bundles. Math. Z.186, 135-142 (1984) · Zbl 0543.32013 · doi:10.1007/BF01215497 [14] Usui, S.: Local Torelli theorem for nonsingular complete intersections. Jap. J. Math.2, 411-418 (1976) · Zbl 0347.14007
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.