Coregularity of Fano varieties
Joaquín Moraga
Abstract
Abstract The absolute regularity of a Fano variety, denoted by $$\hat{\textrm{reg}}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mtext>reg</mml:mtext> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , is the largest dimension of the dual complex of a log Calabi–Yau structure on X . The absolute coregularity is defined to be $$\begin{aligned} \hat{\textrm{coreg}}(X):= \dim X - \hat{\textrm{reg}}(X)-1. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mover> <mml:mtext>coreg</mml:mtext> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>dim</mml:mo> <mml:mi>X</mml:mi> <mml:mo>-</mml:mo> <mml:mover> <mml:mtext>reg</mml:mtext> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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.