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Alexandru Dimca

Publications and source records attributed to Alexandru Dimca.

At least 109 records · Page 6Linked to original sources
arXiv2014

Hilbert series and Lefschetz properties of dimension one almost complete intersections

We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is proved for $n=2$ and a counterexample due to A. Conca is given for such a property when $n=3$. Two conjectures are also stated in the paper.

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arXiv2013

Pencil type line arrangements of low degree: classification and monodromy

The complete classification of (3,3)-nets and of (3,4)-nets with only double and triple points is given. Up to lattice isomorphism, there are exactly 3 effective possibilities in each case, and some of these provide new examples of pencil-type line arrangements. For arrangements consisting of at most 14 lines and having points of multiplicity at most 5, we show that the non-triviality of the monodromy on the first cohomology H^1(F) of the associated Milnor fiber F implies the arrangement is of reduced pencil-type. In particular, the monodromy is determined by the combinatorics in such cases.

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arXiv2013

On the cohomology of the Milnor fibre of a hyperplane arrangement

We investigate the cohomology of the Milnor fibre of a reflection arrangement as a module for the group $Γ$ generated by the reflections, together with the cyclic monodromy. Although we succeed completely only for unitary reflection groups of rank two, we establish some general results which relate the isotypic componenents of the monodromy on the cohomology, to the Hodge structure and to the cohomology degree. Using eigenspace theory for reflection groups, we prove some sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler characteristic, such as the Hodge-Deligne polynomials. We also use monodromy eigenspaces to determine the spectrum in some cases, which in turn throws light on the Hodge structure of the cohomology. These methods enable us to compute the complete story, including the representation of $Γ$ on the Hodge components in each cohomology degree, for some groups of low rank.

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arXiv2013

Koszul complexes and pole order filtrations

We study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial $f$ and the pole order filtration $P$ on the cohomology of the open set $U=\PP^n \setminus D$, with $D$ the hypersurface defined by $f=0$. The relation is expressed by some spectral sequences, which may be used on one hand to determine the filtration $P$ in many cases for curves and surfaces, and on the other hand to obtain information about the syzygies involving the partial derivatives of the polynomial $f$. The case of a nodal hypersurface $D$ is treated in terms of the defects of linear systems of hypersurfaces of various degrees passing through the nodes of $D$. When $D$ is a nodal surface in $\PP^3$, we show that $F^2H^3(U) \ne P^2H^3(U)$ as soon as the degree of $D$ is at least 4.

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arXiv2013

Nonabelian cohomology jump loci from an analytic viewpoint

For a topological space, we investigate its cohomology support loci, sitting inside varieties of (nonabelian) representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its cohomology jump loci, sitting inside varieties of (algebraic) flat connections. We prove that the analytic germs at the origin 1 of representation varieties are determined by the Sullivan 1-minimal model of the space. Under mild finiteness assumptions, we show that, up to a degree $q$, the two types of jump loci have the same analytic germs at the origins, when the space and the algebra have the same $q$-minimal model. We apply this general approach to formal spaces (for which we establish the degeneration of the Farber-Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic jump loci: up to degree $q$, all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic cohomology jump loci, up to degree $q$, into their topological counterpart.

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arXiv2013

Number of Jordan blocks of the maximal size for local monodromies

We prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In case the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.

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arXiv2012

Syzygies of Jacobian ideals and defects of linear systems

Our main result describes the relation between the syzygies involving the first order partial derivatives $f_0,...,f_n$ of a homogeneous polynomial $f\in \C[x_0,...x_n]$ and the defect of the linear systems vanishing on the singular locus subscheme $Σ_f=V(f_0,...,f_n)$ of the hypersurface $D:f=0$ in the complex projective space $\PP^n$, when $D$ has only isolated singularities.

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arXiv2012

Monodromy of triple point line arrangements

We show that the monodromy operator action on the first cohomology group of the Milnor fiber is combinatorially determined for line arrangements with at most triple points and containing at most 18 lines, with one possible exception.

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arXiv2012

Some remarks on limit mixed Hodge structure and spectrum

We give some remarks on limit mixed Hodge structure and spectrum. These are more or less well-known to the specialists, and do not seem to be stated explicitly in the literature. However, they do not seem to be completely trivial to the beginners, and may be worth writing down explicitly.

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arXiv2012

Arithmetic group symmetry and finiteness properties of Torelli groups

We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have finite first Betti number, when the genus is at least four. We also prove that, in this range, the $I$-adic completion of the Alexander invariant is finite-dimensional, and the Kahler property for the Torelli group implies the finite generation of the Johnson kernel.

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arXiv2012

The abelianization of the Johnson kernel

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 $g\ge 4$ and give an explicit presentation of it as a $\Sym H_1(T_g,\C)$-module when $g\ge 6$. 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 $K$ is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of $G$. In this setup, we also obtain a precise nilpotence test.

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arXiv2011

On the syzygies and Alexander polynomials of nodal hypersurfaces

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface. The result gives information on the position of the singularities of a nodal hypersurface expressed in terms of defects or superabundances. The case of Chebyshev hypersurfaces is considered as a test for this result and leads to a potentially infinite family of nodal hypersurfaces having nontrivial Alexander polynomials.

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arXiv2011

Chebyshev curves, free resolutions and rational curve arrangements

First we construct a free resolution for the Milnor (or Jacobian) algebra $M(f)$ of a complex projective Chebyshev plane curve $\CC_d:f=0$ of degree $d$. In particular, this resolution implies that the dimensions of the graded components $M(f)_k$ are constant for $k \geq 2d-3.$ Then we show that the Milnor algebra of a nodal plane curve $C$ has such a behaviour if and only if all the irreducible components of $C$ are rational. For the Chebyshev curves, all of these components are in addition smooth, hence they are lines or conics and explicit factorizations are given in this case.

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arXiv2011

Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements

We investigate the interplay between the monodromy and the Deligne mixed Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of hyperplane arrangement Milnor fibers, we obtain a new result on the possible weights. For line arrangements, we prove in a new way the fact due to Budur and Saito that the spectrum is determined by the weak combinatorial data, and show that such a result fails for the Hodge-Deligne polynomials.

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arXiv2011

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given. It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over $\Q$, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial. We construct a hyperplane arrangement defined over $\Q$, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has not polynomial count. Such examples are shown not to exist in low dimensions.

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arXiv2010

Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations

We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the singular fiber are both rational homology manifolds. Also we introduce quasi-semistable degenerations and calculate the limit mixed Hodge structure by constructing the weight spectral sequence. As a corollary we show non-triviality of the space of vanishing cycles of the Lefschetz pencil associated with a tensor product of any two very ample line bundles except for the case of even-dimensional projective space where two has to be replaced by three.

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arXiv2010

First Milnor cohomology of hyperplane arrangements

We show a combinatorial formula for a lower bound of the dimension of the non-unipotent monodromy part of the first Milnor cohomology of a hyperplane arrangement satisfying some combinatorial conditions. This gives exactly its dimension if a stronger combinatorial condition is satisfied. We also prove a non-combinatorial formula for the dimension of the non-unipotent part of the first Milnor cohomology, which apparently depends on the position of the singular points. The latter generalizes a formula previously obtained by the second named author.

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