## 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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### References:

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