Coregularity of Fano varieties

Abstract

The absolute regularity of a Fano variety, denoted by reg^(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{\ extrm{reg}}(X)$$\\end{document}, is the largest dimension of the dual complex of a log Calabi–Yau structure on X. The absolute coregularity is defined to be coreg^(X):=dimX-reg^(X)-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\hat{\ extrm{coreg}}(X):= \\dim X - \\hat{\ extrm{reg}}(X)-1. \\end{aligned}$$\\end{document}The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of X. In this note, we review the history of Fano varieties, give some examples, survey some theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.