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**Contact**

Institut für Algebra, Zahlentheorie und Diskrete Mathematik (IAZD)

Fakultät für Mathematik und Physik

Leibniz Universität Hannover

Welfengarten 1

30167 Hannover

Phone: +49 511 7623519

Fax: +49 511 7625490

Gebäude 1101

Raum B402

Mail: bailet@math.uni-hannover.de

**Related links**

http://iml.univ-mrs.fr/sgt/gdri-singularities/

My Ph.D. Supervisor

Co-authors

http://math.unice.fr/~dimca/

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:

- Resonance varieties

- Local systems cohomology, see On the Monodromy of Milnor Fibers of Hyperplane Arrangements, Degeneration of Orlik-Solomon algebras and Milnor fibers of complex line arrangements, Homology graph of real arrangements and monodromy of Milnor fiber, A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements.

- Hodge structures, see Tate properties and monodromy of hyperplane arrangements in four variables, On two pencils of cubic curves.

- Minimal CW decompositions of the complement, see Vanishing results for the Aomoto complex of real hyperplane arrangements via minimality.

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.

On two pencils of cubic 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.

Homology graph of real arrangements and monodromy of Milnor fiber

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.

Vanishing results for the Aomoto complex of real hyperplane arrangements via minimality

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.

Degeneration of Orlik-Solomon algebras and Milnor fibers of complex line arrangements

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.

On the Monodromy of the Milnor Fiber of Hyperplane Arrangements

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.

Ph.D.Thesis

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.

Master Thesis

- Generalization of the degeneration homomorphisms in higher dimensions (generalization in higher dimensions of the degeneration homomorphisms introduced in Degeneration of Orlik-Solomon algebras and Milnor fibers of complex line arrangements).

- Summer school "New Perspectives in Hyperplane Arrangements", September 10-14, 2018, Ruhr-Universität Bochum, Germany

- The 5th Franco-Japanese-Vietnamese Symposium on Singularities, 29 Oct.-3 Nov. 2017, Kagoshima University, Kagoshima, Japan
- "Advances in Arrangement Theory" at the Mathematical Congress of the Americas, 24-28 July 2017, Montreal, Canada
- Arrangements and beyond, 6-9 June 2017, CRM Ennio de Giorgi, Pisa, Italy (Talk: A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements)
- Combinatorics of Arrangements of Hyperplanes, 19-21 April, 2017, Leibniz Universität, Hannover, Germany (Talk: Monodromy of Milnor fibers of hyperplane arrangements)
- Colloquium in Linköping, Sweden, December 13, 2016 (Talk: Monodromy of Milnor fibers of hyperplane arrangements)
- Combinatorial Structures in Algebra, Geometry and Topology, Bremen, October 31, 2016 (Talk: Aomoto complexes and monodromy of Milnor fibers of hyperplane arrangements)
- PISA-HOKKAIDO Summer School on mathematics and its applications, Centro De Giorgi, Pisa, 29th August - 16th September 2016 (Talk: Aomoto complexes and monodromy of Milnor fibers of hyperplane arrangements)
- Summer Conference on Hyperplane Arrangements (SCHA) in Sapporo, August 8-12, 2016, Hokkaido University, Japon
- Topology and Geometry of arrangements, Doctoral school and workshop, 6-8 June 2016, University of Fribourg (Talk: Monodromy of pencils of curves)
- Special Session on Topology and Combinatorics of Arrangements (in honor of Mike Falk) at the AMS Spring Eastern sectional meeting, 9-20 March 2016, Stony Brook University, New York (Talk: Vanishing results for the Aomoto complex of real hyperplane arrangements via minimality)
- Singularities and topology, 7-9 March 2016, Nice, France (Talk: Vanishing results for the Aomoto complex of real hyperplane arrangements via minimality)

- Arrangements: topology, combinatorics and stability, 3-6 February 2015, University of Pisa, Italy (Talk: On the monodromy of Milnor fibers of line arrangements with a sharp pair)

- ICERM workshop « Computational Geometric Topology in Arrangement Theory », 6-10 July 2015, Brown University, USA (Talk: Degeneration of Orlik-Solomon Algebras and Milnor fiber of complex line arrangements)

- Problems around hyperplanes arrangements , 18 June 2015, Hokkaido University, Japan (Talk: Minimal Salvetti complex and Milnor fiber of hyperplane arrangements)

- Arrangements of plane curves and related problems, 16-18 March 2015, Tokyo Metropolitan University, Japan (Talk: Degeneration of Orlik-Solomon algebras and Milnor fiber of complex line arrangement)

- First workshop of JSPS-MAE Sakura Program "Geometry and Combinatorics of hyperplane arrangements and related problems", 1-5 September 2014, Hokkaido Universiry, Sapporo, Japan
- Second symposium France-Japon-Vietnam on singularities, 25-30 August 2014, Hokkaido University, Sapporo, Japan (Talk: Tate properties and monodromy of hyperplane arrangements)

- Ph.D. seminar of the ADSFA, 13-15 June 2014, Fréjus, France

- Winter Braids IV (school on algebraic and topological aspects of braid groups), 10-13 February 2014, Dijon, France (Talk: Degeneration of Orlik-Solomon algebras and Milnor fibers of complex line arrangements)

- First symposium France-Japon-Vietnam on singularities, 16-21 September 2013, Nice, France

- Ph.D. seminar of the ADSFA (association of the Ph.D. students of the doctoral school in fundamental and applied sciences belonging to the UNS), 7-9 June 2013, Fréjus, France

- Ph.D. seminar of the doctoral school in fundamental and applied sciences, 23 May 2013, Nice, France

- Research group « singularities and applications » meeting (geometry and topology of complex singularities), 15-19 April 2013, Marseille, France

- Research group « singularities and applications » meeting (characteristic classes), 22-26 October 2012, Porquerolles, France

- Algebraic geometry day Nice – Gênes, 25 June 2012, Nice, France

- Arrangements in Pyrénées, 11-15 June 2012, Pau, France

- Hyperplanes Arrangements and Related Topics, 6 March 2012, Nice, France

- Linköping, Sweden, December 12-19, 2016
- Pisa, Italy, 01-26 February, 2016