Good structures in higher dimensional birational geometry

• Applicant: Professor Dr. Vladimir Lazic
• Subject Area: Mathematics
• Term: from 2013 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 239673722

• Final Report Year: 2020

Final Report Abstract

The aim of the Minimal Model Program is to classify higher dimensional algebraic varieties, generalising the classification of curves and surfaces. The purpose of the classification is to give a rough understanding of the structure of mildly singular projective varieties, and the programme was completely resolved only in dimension 3 in the 1980s. Recently there has been spectacular progress in the field, partially due to me and my coauthors. However, the programme remains far from being complete, and the main open problems left are the Abundance Conjecture and existence of good minimal models. Furthermore, a related Semiampleness Conjecture predicts that a nef line bundle on a variety with Ricci flat curvature has a multiple which is globally generated. During the Emmy Noether programme, I and my collaborators have made significant progress on all of these very important problems. In particular, we made a surprising breakthrough on the Nonvanishing Conjecture, which forms a large part of the Abundance Conjecture. This is the first progress on the conjecture in the last 30 years, and the first general result on the problem in dimensions greater than 3. The result is a mix of techniques from the Minimal Model Program and the recent progress on the stability of the cotangent bundle of projective varieties, together with the latest tools from differential geometry. Furthermore, our methods apply in the context of the Semiampleness Conjecture to give the first general evidence for the conjecture in dimensions greater than 2. This has led us to formulate a vast common generalisation of the Abundance and Semiampleness conjectures, and our work on this generalisation yielded surprising results on the topological nature of semiampleness. For instance, we show that the semiampleness of the canonical class of a projective variety with mild singularities almost always depends only on its first Chern class. As a pleasant side product, we almost completely settle the Abundance Conjecture for pairs whose underlying variety is covered by rational curves. Additionally, we prove several results on the existence of minimal models and the termination of special flips, as well as various cases of a conjecture of Mumford in every dimension, which characterises varieties which contain many rational curves.

Publications

• A note on the abundance conjecture, Algebraic Geometry 2 (2015), no. 4, 476-488.
  T. Dorsch, V. Lazić
  (See online at https://doi.org/10.14231/AG-2015-020)
• Automorphisms of Calabi-Yau threefolds with Picard number three, Higher dimensional algebraic geometry in honour of Professor Yujiro Kawamata’s sixtieth birthday, Adv. Stud. Pure Math., vol. 74, Mathematical Society of Japan, Tokyo, 2017, pp. 279-290
  V. Lazić , K. Oguiso, Th. Peternell
  (See online at https://doi.org/10.2969/aspm/07410279)
• Rationally connected varieties – on a conjecture of Mumford, Sci. China Math. 60 (2017), no. 6, 1019-1028
  V. Lazić , Th. Peternell
  (See online at https://doi.org/10.1007/s11425-016-9039-2)
• Abundance for varieties with many differential forms, Épijournal Géom. Algébrique, Volume 2 (2018), Article Nr. 1
  V. Lazić , Th. Peternell
  (See online at https://doi.org/10.46298/epiga.2018.volume2.3867)
• Abundance for uniruled pairs which are not rationally connected
  V. Lazić
  (See online at https://doi.org/10.48550/arXiv.1908.06945)
• On Nonvanishing for uniruled log canonical pairs
  V. Lazić, F. Meng
  (See online at https://doi.org/10.48550/arXiv.1907.11991)
• On the B-Semiampleness Conjecture, Épijournal Géom. Algébrique, Volume 3 (2019), Article Nr. 12
  E. Floris, V. Lazić
  (See online at https://doi.org/10.46298/epiga.2019.volume3.5063)
• Maps from K-trivial varieties and connectedness problems, Annales Henri Lebesgue 3 (2020), 473-500
  V. Lazić , Th. Peternell
  (See online at https://doi.org/10.5802/ahl.38)
• Nef line bundles on Calabi-Yau threefolds, I, Int. Math. Res. Not. IMRN, Vol. 2020, No. 19, 6070-6119
  V. Lazić , K. Oguiso, Th. Peternell
  (See online at https://doi.org/10.1093/imrn/rny191)
• On Generalised Abundance, I, Publ. Res. Inst. Math. Sci. 56 (2020), no. 2, 353-389
  V. Lazić, Th. Peternell
  (See online at https://doi.org/10.4171/PRIMS/56-2-3)
• On Generalised Abundance, II, Peking Math. J. 3 (2020), no. 1, 1-46
  V. Lazić, Th. Peternell
  (See online at https://doi.org/10.1007/s42543-019-00022-1)
• On the existence of minimal models for log canonical pairs
  V. Lazić, N. Tsakanikas
  (See online at https://doi.org/10.48550/arXiv.1905.05576)
• On the number and boundedness of minimal models of general type, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 5, 1183-1210
  D. Martinelli, S. Schreieder, L. Tasin
  (See online at https://doi.org/10.24033/ASENS.2443)
• Special termination for log canonical pairs
  V. Lazić , J. Moraga, N. Tsakanikas
  (See online at https://doi.org/10.48550/arXiv.2007.06458)