## Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville.(English)Zbl 0659.14007

Let X be a compact, connected Kähler manifold of dimension n, $$Pic^ 0(X)$$ the identity component of the Picard group of X, $$S^ i(X)\subseteq Pic^ 0(X)$$ the analytic subvariety given by $$S^ i(X)=\{L\in Pic^ 0(X)| H^ i(X,L)\neq 0\},$$ $$i\geq 0$$ and $$a: X\to Alb(X)$$ the Albanese map of X. Then $$co\dim (S^ i(X),Pic^ 0(X))\geq \dim(a(X))-i$$. In particular, if $$L\in Pic^ 0(X)$$ is a generic line bundle, then $$H^ i(X,L)=0$$ for $$i<\dim (a(X))$$ (this is a positive answer to some conjectures of Beauville and Catanese).
If X is an irregular surface without irrational pencils then the trivial bundle $${\mathcal O}_ X$$ is an isolated point of $$S^ 1(X)$$ and consequently any (effectively parametrized) irreducible family of curves on X containing at least one canonical divisor has dimension $$\leq p_ g(X)$$ (this gives an upper bound on the dimensions of algebraic deformations of a canonical divisor on X as sought by Enriques). These results are proved by studying the deformation theory of the groups $$H^ i(X,L)$$ as L varies.
Reviewer: D.Popescu

### MathOverflow Questions:

Cohomology of line bundles

### MSC:

 14C22 Picard groups 14C20 Divisors, linear systems, invertible sheaves 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14D15 Formal methods and deformations in algebraic geometry
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### References:

 [1] [ACGH] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Berlin-Heidelberg-New York-Tokyo: Springer 1984 [2] [BS] Banica, C., Stanasila, O.: M?thodes alg?briques dans la theorie des ?spaces complexes. Paris: Gauthier-Villars 1977 [3] [BPV] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berlin-Heidelberg-New York-Tokyo: Springer 1984 · Zbl 0718.14023 [4] [C] Catanese, F.: Moduli of surfaces of general type, in Proceedings of the 1982 conference at Ravello. Lect. Notes Math., vol.997, pp. 90-112. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0517.14011 [5] [CL] Carrell, J., Lieberman, D.: Holomorphic vector fields and K?hler manifolds. Invent. math.21, 303-309 (1973) · Zbl 0258.32013 [6] [D] Deligne, P.: Th?or?me de Lefschetz et crit?res de d?g?n?rescence des suites spectrales. Publ. Math., Inst. Hautes Etud. Sci.35, 259-278 (1968) · Zbl 0159.22501 [7] [E] Enriques, F.: Le Superficie algebriche. Zanichelli 1949 · Zbl 0036.37102 [8] [EV] Esnault, H., Viehweg, E.: Logarithmic De Rham complexes and vanishing theorems. Invent. math.86, 161-194 (1986) · Zbl 0603.32006 [9] [GH] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley and Sons 1978 · Zbl 0408.14001 [10] [K] Koll?r, J.: Vanishing theorems for cohomology groups, to appear in the Proceedings of the 1985 conference at Bowdoin · Zbl 0658.14012 [11] [Mt] Matsumura, H.: Commutative algebra. New York: Benjamin 1970 · Zbl 0211.06501 [12] [M] Mumford, D.: Abelian varieties. Oxford Univ. Press 1970 · Zbl 0223.14022 [13] [U] Ueno, K.: (ed.) Classification of algebraic and analytic manifolds. Progr. Math.39. Birkh?user (1983) · Zbl 0527.14002
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