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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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