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Higher obstructions to deforming cohomology groups of line bundles. (English) Zbl 0735.14004

The set \(\text{Pic}^ 0(M)\) is a complex torus parametrizing isomorphism classes of topologically trivial line bundles on a compact Kähler manifold \(M\). Let \(S^ i_ m(M)=\{{\mathcal U}\in\text{Pic}^ 0(M)\mid h^ i(M,L)\geq m\}\) and \(a_ M:M\to\text{Alb}(M)\) be the Albanese map. The main result shows, for an irreducible component \(Z\) of \(S^ i_ m(M)\), that
(1) \(Z\) is a complex subtorus of \(\text{Pic}^ 0(M)\).
(2) There is an analytic variety \(N\) with dimension less than or equal 1 and an analytic map \(f:M\to N\) with connected fibres such that, for some \(y\), \(Z\subseteq y+f^*(\text{Pic}^ 0(N))\).
(3) For any smooth model \(\widetilde N\) of \(N\), \(\dim(a_ N(\widetilde N))=\dim(N)\).
As a corollary the author shows \(\text{codim}(S^ i(M))\geq\dim(a_ M(M)- i\). The proofs use higher order deformation theory as opposed to an earlier first order theory. Some other ingredients in these proofs are \(\delta\)-operators, Poincaré bundles and the relative Dolbeault complex. — As applications of the above results, the Castelnuovo–De Franchis lemma is generalized and restrictions on the fundamental group of \(M\) obtained. Directions for further work are indicated.

MSC:

14C22 Picard groups
14D15 Formal methods and deformations in algebraic geometry
32J27 Compact Kähler manifolds: generalizations, classification
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[1] A. Beauville, in Ueno , Classification of algebraic and analytic manifolds, Birkhäuser, 1983, pp. 619-620.
[2] Arnaud Beauville, Annulation du \?\textonesuperior et systèmes paracanoniques sur les surfaces, J. Reine Angew. Math. 388 (1988), 149 – 157 (French). · Zbl 0639.14017 · doi:10.1515/crll.1988.388.149
[3] F. Catanese, Moduli of surfaces of general type, Algebraic geometry — open problems (Ravello, 1982) Lecture Notes in Math., vol. 997, Springer, Berlin-New York, 1983, pp. 90 – 112. · Zbl 0517.14011
[4] -, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, preprint. · Zbl 0743.32025
[5] Mark Green and Robert Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389 – 407. · Zbl 0659.14007 · doi:10.1007/BF01388711
[6] Mark Green and Robert Lazarsfeld, A deformation theory for cohomology of analytic vector bundles on Kähler manifolds, with applications, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 416 – 440. · Zbl 0664.32015
[7] William M. Goldman and John J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43 – 96. · Zbl 0678.53059
[8] M. Gromov, Sur le groupe fondamental d’une variété Kählerienne, preprint.
[9] Ziv Ran, On subvarieties of abelian varieties, Invent. Math. 62 (1981), no. 3, 459 – 479. · Zbl 0474.14016 · doi:10.1007/BF01394255
[10] Y. T. Siu, Strong rigidity for Kaehler manifolds and the construction of bounded holomorphic functions, Discrete Groups and Analysis, Birkhäuser, 1981, pp. 44-78.
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