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Syzygies and logarithmic vector fields along plane curves

Alexandru DimcaEdoardo Sernesi — 2014

Journal de l’École polytechnique — Mathématiques

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve C and the stability of the sheaf of logarithmic vector fields along C , the freeness of the divisor C and the Torelli properties of C (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.

Some consequences of perversity of vanishing cycles

Alexandru DimcaMorihiko Saito — 2004

Annales de l’institut Fourier

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.

Which 3-manifold groups are Kähler groups?

Alexandru DimcaAlexander Suciu — 2009

Journal of the European Mathematical Society

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact Kähler manifold, then G must be finite—and thus belongs to the well-known list of finite subgroups of O ( 4 ) , acting freely on S 3 .

Spectrum and multiplier ideals of arbitrary subvarieties

Alexandru DimcaPhilippe MaisonobeMorihiko Saito — 2011

Annales de l’institut Fourier

We introduce a spectrum for arbitrary subvarieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for the coefficients of integral exponents. We show a relation to the graded pieces of the multiplier ideals by using the filtration V of Kashiwara and Malgrange. This implies a partial generalization of a theorem of Budur in the hypersurface case. The key point...

The abelianization of the Johnson kernel

Alexandru DimcaRichard HainStefan Papadima — 2014

Journal of the European Mathematical Society

We prove that the first complex homology of the Johnson subgroup of the Torelli group T g is a non-trivial, unipotent T g -module for all g 4 and give an explicit presentation of it as a S y m . H 1 ( T g , C ) -module when g 6 . We do this by proving that, for a finitely generated group G satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the...

Nonresonance conditions for arrangements

Daniel C. CohenAlexandru DimcaPeter Orlik — 2003

Annales de l’institut Fourier

We prove a vanishing theorem for the cohomology of the complement of a complex hyperplane arrangement with coefficients in a complex local system. This result is compared with other vanishing theorems, and used to study Milnor fibers of line arrangements, and hypersurface arrangements.

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