×

Characteristic varieties and constructible sheaves. (English) Zbl 1139.14009

The main purpose of this paper is to investigate the translated components of the characteristic variety \(V_1(M)\). Following D. Arapura [J. Algebr. Geom. 6, No. 3, 563–597 (1997; Zbl 0923.14010)] such a component \(W\) is described by a pair \((f,\rho)\) where
(a) \(f:M \to S\) is a surjective morphism from the quasi projective smooth surface \(M\) to a smooth curve \(S\), having a connected generic fiber \(F\);
(b) \(\rho\in T(M)\) is a torsion character such that \(W\) is the translate by \(\rho\) of the subtorus \(f^*(T(S))\)
In section 2, basic facts on regular mappings \(f\) and its associated pencils are collected and the notion of admissible map used by Arapura is clarified
In section 3, various properties of irreducible components of characteristic varieties are derived by a careful look at the results by Arapura.
In section 4, the key role played by the constructible sheaves, obtained as direct images of local systems on \(M\) under the mapping \(f\) is emphasized. In particular for a local system \(L\in W\) the dimension of \(H^1(M,L)\) is expressed in terms of the Euler characteristic \(\chi(S)\) and the cardinality of the singular support of the sheaf \(F=R^0f_*L_\rho\).
In the last section, the author associated to \(f\) a finite abelian group \(T(f)\) such that the torsion character \(\rho\) is determined by a character of \(T(f)\). The latter group is computed in terms of the multiplicities of the special fibers of \(f\).

MSC:

14C21 Pencils, nets, webs in algebraic geometry
14F99 (Co)homology theory in algebraic geometry
32S22 Relations with arrangements of hyperplanes
14E05 Rational and birational maps
14H50 Plane and space curves

Citations:

Zbl 0923.14010
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] D. ARAPURA, Geometry of cohomology support loci for local systems. I . J. Algebraic Geom. 6 (1997), 563-597. · Zbl 0923.14010
[2] I. BAUER, Irrational pencils on non-compact algebraic manifolds . Internat. J. Math. 8 (1997), 441-450. · Zbl 0896.14008 · doi:10.1142/S0129167X97000226
[3] A. BEAUVILLE, Annulation du H 1 pour les fibrés en droites plats . In: Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Math. 1507, Springer, Berlin, 1992, 1-15. · Zbl 0792.14006
[4] A. BOREL et al., Intersection Cohomology . Progr. Math. 50, Birkhäuser, 1984.
[5] F. CATANESE, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations . Invent. Math. 104 (1991), 263-289 (with an · Zbl 0743.32025 · doi:10.1007/BF01245076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.