Log canonical rings

ANCS-UEFISCDI grant PN-II-RU-TE-2011-3-0097, Contract number 3 (05/10/2011)

October 2011 - September 2014

Project summary

Graded rings of differential pluri-forms, called log canonical rings, play a key role in the classification theory of algebraic varieties. They are the algebraic counterpart of global and local geometric objects, such as log canonical models and the singularities that appear on them. Great progress has been made recently towards understanding log canonical rings, for example their finite generation. This project aims to sharpen the known methods and to introduce new ones, in order to understand the degrees of the generators of log canonical rings.

Project coordinator

Florin Ambro, Scientific Researcher II at Institute of Mathematics "Simion Stoilow" of the Romanian Academy

Research team

Radu Titiu, Graduate Student

Publications

Ambro, F., An injectivity theorem, Compositio Mathematica 150(6) (2014), 999 - 1023
Ambro, F., Cyclic covers and toroidal embeddings, preprint arXiv:1310.3951 (submitted for publication)

Results obtained

A generalization of the injectivity theorem of Esnault and Viehweg (report)
An improved version of the cyclic covering trick (report)

Research talks

''Division Flow'', Laboratory of algebraic geometry and its applications, Faculty of Mathematics, Higher School of Economics, Moscow, October 14 2011
''Singularities in Birational Geometry'', Symposium on Projective Algebraic Varieties and Moduli, Busan, Korea, February 16 2012
''Singularities in Birational Geometry'', A series of six lectures delivered at Institute of Mathematics, Chinese Academy of Sciences, Beijing, February 20-March 9 2012
''Division Flow'', Colloquim, Institute of Mathematics, Chinese Academy of Sciences, Beijing, March 16 2012
''Toric division flow'', Workshop for Young Researchers in Mathematics, Ovidius University, Constanta, May 10 2012
''Introduction to minimal log discrepancies'', Workshop on ACC for minimal log discrepancies and termination of flips, AIM, Palo Alto USA, May 14 2012
''An injectivity theorem'', Workshop on Algebraic Geometry, Florence, Italy, September 14 2012
''Combinatorial algebraic varieties'', Algebra and Geometry Seminar, Ovidius University, Constanta, November 16 2012
''An injectivity theorem'', One day Workshop on Algebraic Geometry, National Taiwan University, Taipei, May 10 2013
''Singularities in Birational Geometry'', Colloquim, National Taiwan University, Taipei, June 3 2013
''An injectivity theorem'', Special Session on Algebraic Geometry, Joint International Meeting of the AMS and the Romanian Mathematical Society, Alba Iulia, June 28 2013
''An injectivity theorem'', Complex Geometry Conference, Singapore, August 5 2013
''An injectivity theorem'', Anniversary Conference Faculty of Sciences - 150 years, Bucharest, August 31 2013
''Singularities in Birational Geometry'', Geometry and Physics Seminar, Department of Theoretical Physics, IFIN-HH, Magurele, October 25 2013
''Division Flow'', Singularity Theory Seminar, Nihon University, Tokyo, November 11 2013
''An injectivity theorem'', Algebraic Geometry Seminar, University of Tokyo, November 19 2013
''Cyclic covers and toroidal embeddings'', Mini Workshop on Birational Geometry, University of Tokyo, November 21 2013
''An injectivity theorem'', Conference on Positivity, Vanishing Theorems and Applications, Lille, May 12 2014
''On toric log Fano varieties'', Workshop Frontiers of rationality, Longyearbyen, Norway, July 15 2014

Research visits

Workshop for Young Researchers in Mathematics, Ovidius University, Constanta, May 10 - 11 2012, May 22 - 23 2014
Interactions of Birational Geometry with other fields, LMS - EPSRC Durham Symposium, University of Durham UK, July 2 - 6 2012
Kahler Geometry and Quantization, Summer School in Cologne, University of Cologne, Germany, July 16 - 20 2012
International Conference: Experimental and Theoretical Methods in Algebra, Geometry and Topology, Eforie Nord, June 21 - 24, 2013
On Lang and Vojta's Conjectures, Conference in CIRM, Luminy, France, March 3 - 7, 2014
Edge Days, University of Edinburgh, UK, June 6 - 8 2014

Visitors at IMAR

Paolo Cascini (Imperial College London), January 16 -21 2012
Enrica Floris (University of Strasbourg), April 22 - May 12 2012
Jihun Park (Pohang University of Science and Technology), March 8 - 16 2013
Yifei Chen (Chinese Academy of Sciences, Beijing), September 7 - 17 2013
Sung Rak Choi (IBS, Center for Geometry and Physics, Pohang), January 15 - 29 2014
Hara Charalampous (Aristotle University of Thessaloniki), August 31 - September 6 2014
Charles Boyer (University of Albuquerque), September 6 - 14 2014

Workshop at IMAR

Locally Free Geometry Seminar (coorganized with Caucher Birkar, Paolo Cascini and Dima Panov), September 13-14 2013