| Summary: | RESULTS OBTAINED: For each specific goal, describe or summarize the results obtained. Relate each one to work already published and/or manuscripts submitted. In the Annex section include additional information deemed pertinent and relevant to the evaluation process. The maximum length for this section is 5 pages. (Arial or Verdana, font size 10). (1) Geometry of equivariant compactifications of vector groups. The classification of Fano varieties is a central problem in Algebraic Geometry. More precisely, the Minimal Model Program (MMP, for short) essentially predicts that every single algebraic variety can be decomposed into 3 types of varieties (after performing suitable birational modifications): Fano varieties, Calabi-Yau varieties, and canonically polarized varieties. These varieties are algebro-geometric analogues of manifolds with positive, zero, and negative curvature, respectively. By classical results by Kollár-Miyaoka-Mori (1992), in each fixed dimension n there is a finite number of families of smooth Fano manifolds of that given dimension. This result has been recently generalized to mildly singular Fano varieties (BAB Conjecture) by Caucher Birkar (2019, 2021). Although, in principle, we can hope to describe all Fano varieties in each fixed dimension, in practice this is a difficult task and sometimes is more convenient to look for general properties or to add some geometric restrictions that make the classification problem easier. In dimension 1 there only one Fano variety (the projective line), while in dimension 2 there are 10 Fano surfaces, which are called del Pezzo surfaces. In dimension 3, the situation is much more complicated, and the classification results are based on techniques from the Minimal Model Program. More precisely, Iskovskikh and Mori-Mukai classified all smooth Fano threefolds into 105 deformation families at the end of the 70s and the beginning of the 80s, respectively. Later, Fujita and Mukai extended these results to higher dimensional Fano varieties with “high Fano index” ...
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