Loading...
Please wait, while we are loading the content...
Similar Documents
A G ] 8 O ct 2 01 8 Logarithmic Chow theory
| Content Provider | Semantic Scholar |
|---|---|
| Author | Barrott, Lawrence Jack |
| Copyright Year | 2018 |
| Abstract | Log geometry was introduced by Kato in [Igu89] based on ideas of Illusie and Fontaine, who expanded on it in [Ill94] to control degenerations of varieties. This theory enhances a scheme with a sheaf of monoids over the structure sheaf. Many of the notions of algebraic geometry, such as smoothness and flatness, can be copied across to log geometry, as explained most vividly by Olsson in [Ols03]. This led us to ask whether other geometric notions can be transferred to this setting. In this paper we describe a refined Chow theory for log varieties. This is motivated by the construction by Abramovich, Chen, Gross and Siebert of log Gromov-Witten invariants in [GS13]. This produces a dimension graded family of Abelian groups supporting a push-forward and pull-back along proper and log flat morphisms respectively. Chow groups and rings have a central role in many areas of Algebraic Geometry. Classical enumerative problems find their most succinct statements in this language whilst problems in the Minimal Model Program require an intimate knowledge of the positivity of various intersections. The construction by Fulton and MacPherson of Chow cohomology in terms of bivariant theory explained many features of the theory which previously had been mysterious. This work was continued by Kresch to extend the theory to more and more unfamiliar worlds, culminating in a theory for Artin stacks. Log geometry is an extension of algebraic geometry to include degeneration data. Most importantly the notion of smoothness generalises well to this setting, certain log schemes become smooth despite their underlying schemes not being classically smooth. The key concept we exploit here is the categorical definition of a monomorphism. Monomorphisms of log schemes were first studied in work of Mochizuki [Moc15] although the relation to this work is unclear. The driving force behind this paper is that a cycle on a scheme should not be thought of as a closed immersion, but rather as a proper monomorphism. That the two concepts are equivalent for schemes was proven by Grothendieck in [DG67]. In this paper we introduce the relevant concepts and construct the log Chow groups of a log scheme. In future work we will construct the appropriate bivariant theories and construct an intersection pairing for log smooth schemes. An intuitive example is that of P with its toric log structure. The cycles classes here can be thought of as genuine rationally equivalent cycles where we avoid those parts of the rational equivalence where the log structure changes. To begin with the only possible zero cycles are points with trivial log structure. There is precisely one such class, any two trivial points are rationally equivalent. Then there are two possible types of one dimensional cycles, strict maps from the standard log point to the toric fixed points and the entirety of P. The points are rigid since any nearby point has trivial log structure, the whole of P is rigid for dimension reasons. Therefore we find that the log Chow groups are given by A†(P ) ∼= Z and A†(P ) ∼= Z. For now let us introduce a motivating philosophy and explain the striking features it uncovers: There are interesting proper log étale maps generically of degree one. Examples of which include blowups along toric ideals of toric varieties. We call such morphisms log refinements. We will see that such blow ups are monomorphisms of log schemes, despite the fact that geometrically they potentially send whole divisors to a point. In fact we believe that these should produce isomorphisms on any sensible log geometric construction you make. Example 1. We consider A with its toric log structure. Let π : X → A be a weighted blow up of the origin and i : E → X be the strict inclusion of the exceptional curve. Then the composite πi : E → A is a proper monomorphism, the logarithmic generalisation of a closed immersion, and so defines a cycle on A. Another miraculous part of this definition is that the category of log schemes has a form of image factorisation: Theorem 1. Let f : X → Y be a morphism of log schemes. Then there exists refinements f̃ : X̃ → Ỹ lifting f and a factorisation p̃ : X̃ → Z̃, ĩ : Z̃ → Ỹ such that f̃ = ĩp̃ and ĩ is a proper monomorphism. If q̃ : X̃ → W̃ and j̃ : W̃ → Ỹ is another factorisation of f̃ with j̃ a proper monomorphism then there is a proper monomorphism k : Z̃ → W̃ such that ĩ = j̃k. The choice of refinement dissappears as soon as one inverts these refinement morphisms. This work was carried out in part during my PhD studentship in Cambridge, supported by Trinity College, the Cambridge Philosophical Society and the Department for Pure Mathematics and Mathematical Statistics. The remaining work and writing was carried out during a postdoctoral fellowship at NCTS in Taipei. I would like to thank Mark Gross for his insight on this project, and to Helge Ruddat for looking over a draft version and providing comments. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv-export-lb.library.cornell.edu/pdf/1810.03746 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
