How to make log structures

Abstract

We introduce the concept of a viable generically Gorenstein toroidal crossing (ggtc) space

. This generalizes the concept of Gorenstein toroidal crossing scheme, which in turn generalizes that of a simple normal crossing scheme. On such a space

upper Y Y $Y$

, we define a sheaf

script upper L script upper S Subscript upper Y L S Y $\mathcal{LS}_Y$

, intrinsic to

upper Y Y $Y$

, by means of an explicit construction. Our main theorem establishes a bijection between the set

upper L upper S Subscript k Sub Superscript dagger Baseline left parenthesis upper Y right parenthesis LS k † ⁡ ( Y ) $\operatorname {LS}_{k^\dagger } (Y)$

of isomorphism classes of log structures on

upper Y Y $Y$

over the log point

upper S p e c k Superscript dagger Spec ⁡ k † $\operatorname {Spec} k^\dagger$

that are compatible with the ggtc structure and the set

normal upper Gamma left parenthesis upper Y comma script upper L script upper S Subscript upper Y Superscript times Baseline right parenthesis Γ ( Y , L S Y × ) $\Gamma (Y,\mathcal{LS}_Y^\times )$

of nowhere-vanishing global sections of

script upper L script upper S Subscript upper Y L S Y $\mathcal{LS}_Y$

. The definition of

script upper L script upper S Subscript upper Y L S Y $\mathcal{LS}_Y$

by explicit construction permits the effective construction of log structures on

upper Y Y $Y$

; it also enables logarithmic birational geometry , in particular the construction, in some cases, of resolutions of singular log structures. Our work generalizes [Gross and Siebert, J. Differential Geom. 72 (2006), 169–338, Theorem 3.22], adapting the original proof with techniques from the theory of

2 2 $2$

-groups and local line bundle systems.