Resumen de Regularity in logarithmic algebraic geometry: a different viewpoint

Jesús Conde Lago

• A fundamental concept in algebraic geometry is nonsingularity. In general, it is much easier to work with nonsingular schemes than with singular schemes. A very useful tool is the resolution of singularities, shown by Hironaka [H]: given a variety X over a field of characteristic zero, there is a birational morphism from Y to X, where Y is a nonsingular variety. This morphism is also an isomorphism on the open set of nonsingular points in X. For arbitrary characteristic, we have the result of de Jong [dJ]: given a variety X over a field, there is a proper surjective morphism from Y to X which is generically finite with Y nonsingular. Logarithmic algebraic geometry is another concept which help us deal with some singular varieties. This theory was introduced by Fontaine and Illusie, and mainly developed by Kato [K1, K2, HK, K3, KN, KKN]. On the one hand, logarithmic algebraic geometry includes algebraic geometry in the sense that all scheme is a logarithmic scheme (with trivial logarithmic structure). On the other hand, certain singular schemes can be provided with a suitable logarithmic structure for the purpose of achieving a behaviour similar to those nonsingular schemes. That allows an easier study within the logarithmic context. Toric varieties are an important example in this context [K2]. Besides, logarithmic theory maintains good relationship with de Jong’s theory of alterations. The study of logarithmic singularities in algebraic geometry started by Kato ([K1] and specially [K2]) and described above, although it is very important, it is also incomplete and does not answer many questions that are usually asked. We intend to continue and deepen the study of logarithmic singularities. [H] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109-326. [HK] Hyodo, O.; Kato, K. Semi-stable reduction and crystalline cohomology with logarithmic poles. Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque No. 223 (1994), 221-268. [dJ] de Jong, A.J. Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math., 83 (1996), 51-93. [K1] Kato, K. Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD (1989), 191-224. [K2] Kato, K. Toric singularities. Amer. J. Math. 116 (1994), no. 5, 1073-1099. [K3] Kato, K. Semi-stable reduction and p-adic étale cohomology. Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque No. 223 (1994),269-293. [KKN] Kajiwara, T.; Kato, K.; Nakayama, C. Logarithmic abelian varieties, Part IV: Proper models. Nagoya Math. J. 219 (2015), 9-63. [KN] Kato, K.; Nakayama, C. Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C. Kodai Math. J. 22 (1999), no. 2, 161-186.