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.


14C22 Picard groups
14D15 Formal methods and deformations in algebraic geometry
32J27 Compact Kähler manifolds: generalizations, classification
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