I am working in Algebraic Geometry
My research interests: Topology of complex algebraic varieties, Singularities of complex algebraic varieties, Sheaves theory, Hodge structure, Hyperplane arrangements, Combinatorics, Tropical geometry
Institut für Algebra, Zahlentheorie und Diskrete Mathematik (IAZD)
Fakultät für Mathematik und Physik
Leibniz Universität Hannover
Phone: +49 511 7623519
Fax: +49 511 7625490
My Ph.D. Supervisor
An hyperplane arrangement is a finite set of hyperplanes in a projective or affine space. Hyperplane arrangements are in the center of many researches on the crossroads of algebra, geometry, topology and combinatorics, since the works of V.I. Arnold, E. Brieskorn, P. Deligne, P. Orlik and L. Solomon.
This theory aims to understand the extent to which inherent combinatorial data determines geometric or topological invariants of arrangements. For instance, we know for a long time thanks to Orlik and Solomon that the cohomology groups of the complements of hyperplane arrangements are completely determined by the combinatorics. One can ask if it is the same for the cohomology groups of the Milnor fiber, and approach the question by considering the action of the monodromy.
Are the monodromy operators, or at least the Betti numbers of the Milnor fiber, determined by the arrangements combinatorics?
This intriguing open question is related to many things, among other:
We give a general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems. We also apply this result in order to determine the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements.
We give two examples of curve arrangements of pencil type in the projective plane, which are very close to line arrangements, though the action of the monodromy on the first cohomology group of the Milnor Fiber has eigenvalues of order 5 and 6, showing that surprising situations can occur for larger classes of curve arrangements than for line arrangements. Our computations rely on an algorithm given by A. Dimca and G. Sticlaru which detects the non trivial monodromy eigenspaces of free curves.
We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the corresponding deconing arrangement and its boundary map. We describe an algorithm which computes possible eigenvalues of the first monodromy operator. We prove that, if a condition on some intersection points of lines in the deconing arrangement is satisfied, then the only possible non trivial eigenvalues are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.
We prove vanishing results for the cohomology groups of the Aomoto complex over an arbitrary coefficient ring for real hyperplane arrangements. The proof uses the minimality of arrangements and descriptions of the Aomoto complex in terms of chambers. Our methods are used to present a new proof for the vanishing theorem of local system cohomology groups, a result first proved by Cohen, Dimca, and Orlik.
We give a vanishing theorem for the monodromy eigenspaces of the Milnor fibers of complex line arrangements such that there exists a line containing exactly one point with multiplicity greater or equal to three. By applying the result of Papadima- Suciu, our theorem is deduced from the vanishing of the cohomology of certain Aomoto complex over finite fields. In order to prove this, we introduce degeneration homomorphisms of Orlik-Solomon algebras.
We construct a new graph by using the information contained in the intersection lattice of arrangements. By using this graph, the modular bound of the local system cohomology groups given by Papadima- Suciu and vanishing results of Cohen-Dimca-Orlik, we give a large class of arrangements such that the action of the monodromy on the first cohomology of the Milnor fiber is the identity.
My phd thesis gives vanishing results for the non trivial eigenspaces of the monodromy on the Milnor fiber.
The first one uses a new graph constructed with the information contained in the intersection lattice of the arrangements, whose connectivity implies the nullity of the cohomology of a certain Aomoto complex.
The second result applies to projective line arrangements such that there exists a line containing exactly one point with multiplicity greater or equal to three (joint work with M. Yoshinaga).
Finally we consider mixted Hodge structure of the cohomology of the Milnor fiber and give an equivalence between triviality of the monodromy, Tate property, and the nullity of the non integer spectrum’s coefficients for central and essential arrangements in the complex space of dimension four.
My Thesis defense
My master’s thesis introduces the basic objects of hyperplane arrangements theory (intersection lattice, characteristic polynomial…). We give a new proof of a classical formula due to Roberts (1889). Then we consider finite fields and the cardinality of the Milnor fiber (related to polynomial count properties of the Milnor fiber). We compute this cardinal in special cases.