Project PN II IDEI nr. 502/2009 code ID_2228

 

Director: Prof. dr. Razvan Dinu LITCANU

  



  Project title

  Research Team

  Objectives

  Publications


1. PROJECT TITLE

Project's Title : COHOMOLOGICAL METHODS IN ALGEBRAIC GEOMETRY AND COMPLEX GEOMETRY
Code: ID_2228

1.1 Thematic fields:

11 Main Sciences: Mathematics

Sub-field: Algebra, mathematical logic and number theory / Geometry, topology and global analysis

 

1.2 Abstract:

In the last decades algebraic geometry became one of the leading fields in mathematics. The main goal of this project is to develop cohomological methods in order to study complex objects in algebraic geometry and complex geometry. Motivated by the increasing amount of applications of algebraic geometry in cryptography and coding theory, we are particularly interested in methods having explicit, combinatorial and even computational nature.

 

The most relevant research directions that we intend to include in this project are: an axiomatic theory of singular Bott-Chern cohomology classes; the relations between the geometry of projective varieties and the syzygy spaces, via the Koszul cohomology; the geometry of biharmonic submanifolds in the complex projective space and classification results; obtainment of functional and/or combinatorial relations between hyperlogarithms; numerical characterizions for line bundles on algebraic varieties, which induce morphisms to projective spaces or embeddings in projective spaces. In this context we intend to attack several problems of great interest in algebraic geometry: arithmetic Grothendieck-Riemann-Roch type results in Arakelov theory, Green and Green-Lazarsfeld conjectures, Fujita΄s numerical estimates concerning ample line bundles on projective varieties, the variation of mixed Hodge structures. During the realization of this project we shall continue and consolidate international scientific cooperation and initiate new collaborations.

 

2. RESEARCH TEAM

 No. 

 Family and given names 

 Birth year 

 Academic / Scientific title 

 PhD 

 1

 APRODU MARIAN

 1970

 CS I

 Yes

 2

 ONICIUC DUMITRU CEZAR

 1972

 Associate Professor

 Yes

 3

 AMBRO FLORIN

 1972

 CS II

 Yes

 4

 DAN NICUSOR DANIEL

 1969

 CS III

 Yes

 5

 CUZUB STEFAN ANDREI*

 1987

 Assistant Researcher

 Graduate

  student

* Ştefan Andrei Cuzub has been a member of the research team since October 1, 2009; until September 30, 2009 Alexandru-Petre Tache, graduate student at the University of Bucharest, was a member of the team.



3. OBJECTIVES

The objectives for the year 2009, intermediate stage, are the following:

  1. Axiomatic characterization of singular Bott-Chern currents and their cohomology classes;
  2. Study and research visits.

 

The objectives for the year 2009, final stage, are the following:

  1. The study of the geometry of syzygies on curves;
  2. Study and research visits.

 

The objectives for the year 2010 are the following:

  1. The study of the combinatorial relations between hyperlogarithms of given weight and level;
  2. Characterization of fiber bundles on projective varieties, that define morphisms in the projective space;
  3. Solving some cohomological problems on curves;
  4. The study of the relations between the geometry of the projective varieties and the syzygies;
  5. Development of the general theory of hyperlogarithms;
  6. Study and research visits.

 

The objectives for the year 2011 are the following:

  1. Obtaining results of Grothendieck-Riemann-Roch type in Arakelov theory;
  2. The study of biharmonic submanifolds in the complex projective space CP^n;
  3. Characterization of fiber bundles on projective varieties, that define morphisms in the projective space;
  4. Study and research visits.



4. PUBLICATIONS
The objectives for 2009 and 2010 have been accomplished. The results obtained in this research project have been published as follows.

1.      A. Balmuş, C. Oniciuc: Biharmonic Surfaces of S^4, Kyushu J.of Mathematics 63, No. 2 (2009), 339-345 (revista ISI)

2.      M. Aprodu, D. Naie: Enriques diagrams and log-canonical threshold for curves on smooth surfaces, Geometriae Dedicata 146 (2010), 43-66 (revista ISI), DOI 10.1007/s10711-009-9425-7

3.      J.I. Burgos Gil, R. Litcanu : Singular Bott-Chern classes and the Grothendieck-Riemann-Roch theorem for closed immersions, Doc. Math. 15 (2010), 73—176  (revista ISI)

4.      D. Fetcu, S. Montaldo, E. Loubeau, C. Oniciuc: Biharmonic submanifolds in CP^n,  Math. Zeitschrift 266 (2010), 505-531   (revista ISI)

5.      A. Balmus, D. Fetcu, C. Oniciuc: Harmonic and biharmonic maps at Iasi, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 56 (2010), 81-96

6.      D. Fetcu, C. Oniciuc: A note on integral C-parallel submanifolds in S^7(c). To appear in Revista de la Union Matematica Argentina

7.      M. Aprodu, G. Farkas: Koszul Cohomology and Applications to Moduli. To appear in “Aspects of vector bundles and moduli” Clay Mathematical Institute Proc., AMS (26 pg.)

8.      F. Ambro: On the classification of toric singularities, Combinatorial Commutative Algebra and Computer Algebra , V. Ene and E. Miller (Ed.), Contemporary Mathematics 502, 2009, 1-4

9.      J.I. Burgos Gil, G. Freixas, R. Litcanu: Some recent results on generalized analytic torsion classes. To appear in “Al. Myller” Mathematical Seminar Centennial Conference Proceedings, AIP Conference Proceedings Series

10.  A. Balmus, S. Montaldo, C. Oniciuc: Properties of biharmonic submanifolds in spheres, J. Geom. Symmetry Phys. 17 (2010), 87-102

11.  P. Baird, E. Loubeau, C. Oniciuc : Harmonic and biharmonic maps from surfaces. To appear in Proceedings of the Cagliari Conference (2009), Contemporary Mathematics.

12.  J.I. Burgos Gil, G. Freixas, R. Litcanu: Generalized holomorphic analytic torsion, arxiv:1011.3702, 2010 (submitted)

13.  N. Dan: Sur la conjecture de Zagier pour n=4, II  (submitted)

14.  F. Ambro: Basic properties of log canonical centers, to appear in Proceedings of the Conference “Classification of Varieties”, Schiermonnikoog Island, Ed.: C. Faber et al. (to be published by EMS)