Logarithmically smooth deformations of strict normal crossing logarithmically symplectic varieties

Transcription

1 Logarithmically smooth deormations o strict normal crossing logarithmically symplectic varieties Dissertation zur Erlangung des Grades Doktor der Naturwissenschaten am Fachbereich 08 Physik, Mathematik und Inormatik der Johannes Gutenberg-Universität in Mainz, vorgelegt von Daniel André Hanke, geb. in Mainz Mainz, den 9. Februar 2015

2 Datum der mündlichen Prüung: 20. März 2015 D77

3 i Abstract In this thesis we give a deinition o the term logarithmically symplectic variety; to be precise, we distinguish even two types o such varieties. The general type is a triple (, comprising a log smooth morphism : X Spec κ o log schemes together with a lat log connection : L Ω 1 L and a (, ω) -closed) log symplectic orm ω Γ (X, Ω 2 L). We deine the unctor o log Artin rings o log smooth deormations o such varieties (,, ω) and calculate its obstruction theory, which turns out to be given by the vector spaces H i (X, B (, )(ω)), i = 0, 1, 2. Here B (, )(ω) is the class o a certain complex o O X -modules in the derived category D(X/κ) associated to the log symplectic orm ω. The main results state that under certain conditions a log symplectic variety can, by a lat deormation, be smoothed to a symplectic variety in the usual sense. This may provide a new approach to the construction o new examples o irreducible symplectic maniolds. In dieser Arbeit geben wir eine Deinition des Terms logarithmisch-symplektische Varietät; genau genommen unterscheiden wir sogar zwei Typen solcher Varietäten. Der allgemeinere Typ ist dabei ein Tripel (,, ω) bestehend aus einem log glatten Morphismus : X Spec κ von log Schemata, zusammen mit einem lachen log Zusammenhang : L Ω 1 L und einer (bezüglich geschlossenen) log symplektischen Form ω Γ (X, Ω 2 L). Wir deinieren den Funktor von log Artinringen der Deormationen solcher Varietäten (,, ω) und berechnen dessen Hindernistheorie, die sich als durch die Vektorräume H i (X, B (, )(ω)), i = 0, 1, 2, gegeben herausstellt. Dabei ist B (, )(ω) die Klasse eines gewissen Komplexes von O X -Moduln in der derivierten Kategorie D(X/κ), der zur log symplektischen Form ω gehört. Die Hauptresultate sagen aus, dass sich unter gewissen Voraussetzungen eine log symplektische Varietät mittels einer lachen Deormation zu einer symplektischen Varietät im üblichen Sinne glätten lässt. Dies lieert möglicherweise einen neuen Ansatz ür die Konstruktion neuer Beispiele irreduzibler symplektischer Mannigaltigkeiten. Dans cette thèse nous donnons la deinition du terme variété logarithmiquement symplectique; par souci d exactitude, nous distinguons même deux types de telles variétés. Le type général est un triplet (,, ω) qui se compose d un morphisme log lisse : X Spec κ de schémas log avec une connexion log plate : L Ω 1 et une orme log symplectique ( -ermée) ω Γ (X, Ω 2 L). Nous déinissons le oncteur des anneaux artiniens des déormations log lisses de telles varietés (,, ω) et calculons sa théorie d obstruction, qui se trouve d être donnée par les espaces vectoriels H i (X, B (, )(ω)), i = 0, 1, 2. Sachant que B (, ) (ω) est la classe d un certain complexe des O X-modules dans la catégorie derivée D(X/κ) associée à la orme log symplectique ω. Les résultats principaux établissent que sous certain conditions une variété log symplectique admet une déormation plate dont la

4 ii ibre générale est une variété symplectique lisse au sens usuel. Ceci apporte potentiellement une autre approche pour la construction de nouveaux exemples de variétés symplectiques irréductibles. Introduction Compact hyperkähler maniolds these are compact Kähler maniolds X the holonomy group o which is the symplectic group Sp(dim X) stand in the ocus o investigations by complex algebraic geometry not only, but just since Bogomolov s decomposition theorem: Theorem (De Rham, Berger, Bogomolov, Beauville, c. [1, Thm. 1(2)]) Let X be a compact Kähler maniold with zero Ricci curvature. Then there exists a inite étale cover : X X such that X = T V i X j, where T is a complex torus, where the V i are compact Calabi-Yau maniolds (these are compact simply connected Kähler maniolds o dimension m i 3 with holonomy group SU(m i )) and where the X j are compact simply connected hyperkähler maniolds. In the language o algebraic geometry, compact hyperkähler maniolds correspond to irreducible symplectic varieties which are proper over Spec C; these are deined as ollows: A symplectic variety is a smooth variety X over Spec C which possesses a global 2-orm ω Γ (X, Ω 2 X/C ), the so-called symplectic orm, such that the associated O X-linear map T X/C Ω 1 X/C is an isomorphism; equivalently, such that its associated skew-symmetric pairing T X/C OX T X/C O X is non-degenerate. This implies that the dimension o X is even. An irreducible symplectic variety is a symplectic variety X the symplectic orm ω o which generates the ring H 0 (X, Ω X/C ) as an H0 (X, O X )-algebra. In particular, such a variety is simply-connected. The decomposition theorem then takes the ollowing orm: Theorem (Beauville, c. [1, Thm. 2(2)]) Let X be a Kähler variety which is proper and smooth over Spec C and the irst Chern class o which is zero. Then there exists a inite étale cover : X X such that X = T V i X j, where T is a complex torus, where the V i are projective Calabi-Yau varieties (these are simply connected varieties o dimension m i 3, smooth over Spec C, with trivial canonical line bundle and such that H 0 (V i, Ω p V ) = 0 or 0 < p < m i/c i) and where X j is a proper irreducible symplectic Kähler variety.

5 iii Both proper symplectic varieties and proper Calabi-Yau varieties are subjects o a rich and prospering research. In contrast to numerous existent examples o proper Calabi-Yau varieties, there are only ew examples o proper symplectic varieties known. All examples known to this day are subsequently listed up to deormation equivalence: a) Hilbert schemes o points on a K3 surace, Hilb n (S) (Beauville, [1, Thm. 3]), o dimension 2n, with b 2 = 23, n N; b) Kummer varieties associated to an Abelian surace, K n (A) (Beauville, [1, Thm. 4]), o dimension 2n, with b 2 = 7, n N; c) M4 (O Grady, [27, 2.0.2]), o dimension 10, with b 2 = 24; d) M (O Grady, [28, 1.4]), o dimension 6, with b2 = 8; where the last two exceptional examples (exceptional, because they do not belong to series like the other examples) were constructed by K. O Grady as resolutions o singular moduli spaces o sheaves on a projective K3 surace. Indeed, all o the above examples may be realised as moduli spaces (or resolutions o those) o sheaves on a K3 surace. However, the method o O Grady used to construct his exceptional examples as resolutions o particular singular moduli spaces ails in all other cases o singular moduli spaces, as shown by D. Kaledin, M. Lehn and C. Sorger in [15]. One method to construct compact Calabi-Yau maniolds was introduced by Y. Kawamata and Y. Namikawa in their paper Logarithmic deormations o normal crossing varieties and smoothing o degenerate Calabi-Yau varieties ([21]). For that purpose one regards (complex analytic) strict normal crossing varieties equipped with a particular logarithmic structure Theorem (c. [21, 4.2]) Let X be a compact Kähler strict normal crossing variety equipped with the log structure o semi-stable type o dimension d 3 and X ν X the normalisation o X. Assume the ollowing conditions: a) ω X = OX ; b) H d 1 (X, O X ) = 0; c) H d 2 (X ν, O X ν ) = 0. Then X is smoothable by a lat deormation. The idea o this method is thus to pass rom the category o Calabi-Yau maniolds to the larger category o strict normal crossing Calabi-Yau varieties to regard deormations o such varieties the general ibre o which is a Calabi-Yau maniold in the original sense.

6 iv Based on a proposal by our advisor M. Lehn, an adaptation o this approach is the core o this thesis: It is known that the ibres o a deormation o a compact hyperkähler maniold over a small analytic disc are again compact hyperkähler maniolds (c. [1, 9]). Yet it seems possible, as in the case o Calabi-Yau maniolds, to start with a singular variety and to smoothen it to receive a proper symplectic variety, in principle. As a irst step, we pass rom algebraic geometry to logarithmic algebraic geometry, i. e. rom the category o schemes to the category o logarithmic schemes (X, α X ); these are schemes X carrying a logarithmic structure α X. Particularly, the notion o smoothness o a morphism o schemes is replaced with the wider notion o logarithmic smoothness o a morphism o log schemes. This yields that any semi-stable strict normal crossing variety is log smooth over the point Spec C as soon as both involved schemes are regarded as log schemes equipped with particular log structures. In this thesis we introduce the notion o a (logarithmically smooth) logarithmically symplectic scheme and particularly o a logarithmically symplectic variety which applies to the case o strict normal crossing varieties over Spec C. In doing so, we distinguish between two types o logarithmically symplectic schemes: Such o non-twisted type and such o generally twisted type. In our main results, in a way similar to that o Y. Kawamata and Y. Namikawa, we give conditions under which a logarithmically symplectic variety o the respective type deorms latly into a smooth symplectic variety in the usual sense, that is, under which it is smoothable. Content and Structure In the irst chapter we introduce the basic deinitions o logarithmic geometry (especially or a reader not too amiliar with that topic) such as the notion o log schemes and their morphisms, charts o log structures and morphisms, and log smoothness as well as basic results about these topics. All these notions and acts are taken rom original articles on the topic by K. Kato and F. Kato, as well as rom the excellent lecture notes on logarithmic geometry by A. Ogus, which have come to be known as the Log book, but which are up to now available only directly rom the author. The second chapter recalls the notion o a unctor o Artin rings in the sense o M. Schlessinger beore introducing the analogously deined unctors o log Artin rings. It then collects results rom the theory o unctors o log Artin rings as established by F. Kato, including the log Schlessinger conditions LH 1 -LH 4 or the existence o a hull or even a universal element, and introduces the notion o log smooth deormations. Be aware, that the content o these irst two chapters is not original, but that it is included in this thesis rom the aore mentioned sources or the convenience o the reader. Having chapters one and two as a basis, chapter three deines step by step the log schemes

7 v with additional data this thesis is working with, namely log schemes with line bundles, log schemes with lat log connections (o rank one) and, eventually, log symplectic schemes. Here it seems reasonable to distinguish between two kinds o such schemes, namely the log symplectic schemes o non-twisted type and o general type, as deined in this chapter. This chapter also introduces coherent sheaves and complexes o coherent sheaves associated to the additional data on the respective log scheme. These are the log Atiyah module o a line bundle, the log Atiyah complex o a lat log connection, and the T -complex and B-complex associated to a log symplectic scheme o non-twisted type and o general type, respectively. These sheaves and complexes o sheaves will deliver the obstruction theory o log smooth deormations o the respective kind o object in chapter our. The middle o the third chapter is taken up by a discussion o the here deined notion o logarithmic Cartier divisors, which was inspired by a section in the lecture notes by A. Ogus, as ar as necessary or this thesis. The chapter ends by recalling special log structures associated to schemes with certain additional data, such as the canonical log structure o strict normal crossing schemes o semi-stable type over Spec k rom A. Ogus lecture notes and F. Kato s work. Again, in this last section o chapter three, nothing is original. In the ourth chapter we deine a collection o deormation unctors or the various objects deined in chapter three. Ater recalling and partly reenacting the log smooth deormation theory o log smooth schemes o F. Kato, we calculate (using Čech-(hyper-)cohomology) the obstruction theory o log smooth deormations o log schemes with line bundles, log schemes with lat log connections and log symplectic schemes. These obstruction theories are given by the (hyper-)cohomology groups H 0, H 1 and H 2 o the sheaves and complexes o sheaves constructed in chapter three. At the end o this chapter we show that any o the beorehand deined deormation unctors possesses a hull in the sense o Schlessinger and that some o them are even prorepresentable (but, as expected, not the unctor o log smooth deormations o log symplectic schemes). Chapter ive contains the main results o this thesis. It begins with technical calculations and results proved or later use in this chapter. By introducing a logarithmic version o the T1 liting principle as introduced by Z. Ran and Y. Kawamata and expanded by B. Fantechi and M. Manetti and by ollowing the techniques o Y. Namikawa and Y. Kawamata including an adaptation o a result o J. Steenbrink, we are able to prove that under certain not to rigid conditions the obstructions given in chapter our vanish and that a log symplectic scheme may be deormed latly into a smooth symplectic scheme in the usual sense. Chapter six collects known examples o log symplectic schemes in our sense, appearing naturally; two o them having been investigated by Nagai. It does however not provide an application o our main theorems which produces new examples o symplectic varieties. The last Chapter deals with questions that could not be satisactorily answered within the scope o this thesis.

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9 Contents 1 Logarithmic Geometry Logarithmic Structures Logarithmic Structures Charts and Coherence Logarithmic schemes Logarithmic schemes Morphisms o log Schemes and Charts Log derivations and log dierentials Logarithmic ininitesimal thickenings Logarithmic smoothness Log latness and log integrality Log smooth deormation Theory Functors o Artin rings Artin rings Functors o Artin rings Obstruction theory o unctors o Artin rings Functors o log Artin rings Log Artin rings Functors o log Artin rings Log smooth deormations Additional Data Line bundles Action o log derivations The log Atiyah module o a line bundle L-Derivations Log Schemes with lat log connection The log Atiyah complex o a lat log connection The Lie derivative o a lat log connection The logarithmic Lie derivative Derivations vii

10 viii 3.3 Log symplectic schemes Non-twisted type General type The T -complex The B-complex Log Cartier divisors Log Cartier divisors and line bundles Log Cartier divisors and lat log connections Log structures associated to other structures The compactiying log structure on schemes with open immersion The Log structure associated to Deligne-Faltings schemes Strict normal crossing schemes and logarithmic structures o semistable type Log Cartier divisors on SNC log varieties Regular varieties with SNC divisors SNC log varieties Log symplectic deormation theory Deormation unctors The log symplectic deormation unctor and related unctors The local Picard unctor and related unctors Obstruction theory o log symplectic deormations Log smooth deormations o log schemes Deormations o log symplectic schemes o non-twisted type Log smooth deormations o log schemes with line bundle Log smooth deormations o log schemes with lat log connection Deormations o log symplectic schemes o general type Overview over the tangent and small obstruction spaces Log symplectic deormations over the standard log point Existence o hulls and pro-representability

11 ix 5 Smoothing o SNC log symplectic schemes The Poincaré residue map or SNC log varieties The Poincaré residue map The normalisation residue map or SNC varieties The log T1 liting principle The reeness o the obstruction spaces over the A n Vanishing o the obstructions and main results The obstruction o log smooth litings Smoothing o non-twisted log symplectic varieties Smoothing o twisted log symplectic varieties Examples Blow-up o a log scheme Examples constructed by blowing up First example: Blowing up a point Second example: Blowing up a Lagrangian Examples o Nagai Preparation Nagai s examples Open questions and outlook 169 A Appendix: Monoids 173 A.1 Monoids A.2 Monoid modules A.3 Ideals and aces A.4 Monoid algebras and toric aine schemes A.5 Monoid homomorphisms Bibliography 181 Glossary 185 Lebenslau 189 Danksagung 191

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13 1 Logarithmic Geometry In this irst chapter we collect the basic deinitions and results o logarithmic geometry, mostly or the convenience o the reader who has not yet had much contact with this topic. The reerences or all results are Logarithmic structures o Fontaine-Illusie by K. Kato ([19]), Log smooth deormation theory by F. Kato ([17]) and notably the yet uninished Lectures on Logarithmic Algebraic Geometry by A. Ogus ([29]), which is an excellent introduction to logarithmic algebraic geometry, available on the authors web page. For the basic deinition and results concerning monoids, see Appendix A. All monoids regarded are commutative. As it is custom in the context o logarithmic geometry we will usually abbreviate the words logarithmic and logarithmically to log. A scheme X may be regarded both as X zar equipped with its classical Zariski-topology and Xét with its étale topology, which is iner due to the act that open immersions are étale. We will draw a distinction between these two possible topologies only i necessary. When writing about an open/étale neighbourhood, we mean an open neighbourhood or an étale neighbourhood, depending on the chosen topology. By a point o a scheme, we mean a geometric point. Since the topology on Xét is iner than on X zar, there is a canonical continuous map υ X : Xét X zar which is the identity as a map on sets, when interpreting Zariski-open subschemes U X as open immersions j U : U X, which are always étale. I F is a shea on Xét, then its restriction to open immersions is given as the shea υ X F : U Γ (j U, F). On the other hand, i F is a shea on X zar, then υ 1 X F : (e: U X) Γ (U, e 1 F) is a shea on Xét. It is clear that υ X υ 1 X F = F. 1.1 Logarithmic Structures The category o sheaves o commutative monoids on a scheme X is denoted by Mon X. We regard the structure shea O X o X as an element o Mon X, always with respect to its multiplication. For a monoid P we will denote by P X its constant shea on X, which is the shea associated to the preshea taking every open subset U (respectively, every étale morphism U X) to P. For a shea o monoids M, speciying a monoid homomorphism P Γ (X, M) is equivalent to giving a morphism o sheaves o monoids P X M Logarithmic Structures Deinition ([19, 1.1,1.2],[29, III.1.1.1]) Let X be a scheme. A prelogarithmic structure on X is a morphism o sheaves o monoids α: M O X. It is called a logarithmic structure i it is a logarithmic morphism o sheaves o monoids, i. e. i the restricted morphism α 1 (O X ) O X is an isomorphism. Due to the act, that or 1

14 2 CHAPTER 1. LOGARITHMIC GEOMETRY any log structure we have α 1 (O X ) = M, we identiy O X with the subshea o units M o M via this isomorphism. I α is a (pre)log structure, we will reer to the shea o monoids M as the (pre)logarithmic structure shea o α (or o (X, α)) and denote it by M α. A morphism ϕ: β α o prelog structures is a morphism o sheaves o monoids ϕ: M β M α such that α ϕ = β. A morphism ϕ: β α o corresponding log structures is a morphism o prelog structures. We write prelog X and Log X or the category o prelog structures and log structures on X, respectively. The initial object in Log X is the inclusion ι: O X O X, called the trivial log structure on X. The initial object o prelog X is the inclusion 1: 1 O X. The inal object in both categories is the identity ιd: O X O X, called the hollow log structure on X (c. [29, III.1.1.3]). In a log structure α, the shea o monoids M α may be written additively or multiplicatively. We will use the multiplicative notation mostly. However, when written additively, a log structure α should be thought o as a shea M α o logarithms o certain regular unctions together with an exponential map M α O X given by α. The set o logarithms o a unction is then α 1 (), which might also be empty, and the logarithm o a unit is unique (cp. [29, III.1.1.2]). For example, the trivial log structure ι on Spec C can be written either multiplicatively as the inclusion C C or additively as the well-deined exponential map C/(2πiZ) C, m exp(m), since indeed exp: (C/(2πiZ), +) (C, ) is an isomorphism with (welldeined) inverse map log. For the deinition o the quotient o a monoid by a subgroup see Appendix A Deinition Let α be a log structure on X. The quotient shea M α = M α /α 1 (O X ) = M α/m α is called the characteristic monoid shea o α. Analogously, i ϕ: β α is a morphism o log structures β : M β O X and α: M α O X, we associate a shea o monoids M ϕ := M α/β := M α /ϕ(m β ) which we call the relative characteristic shea o ϕ or o α over β. A morphism ϕ: β α o log structures induces a homomorphism o monoid sheaves ϕ: M β M α. The morphism ϕ an isomorphism i and only i ϕ is. We have M ϕ = M α /ϕ(m β ). Given a diagram o monoid sheaves (or prelog structures) N M N on X we write N M N or its pushout, which is the shea associated to the preshea o monoids U N (U) M(U) N (U). I M is the trivial shea o monoids, we write N N instead o N 1 N (compare to the conventions or monoids in Appendix A).

15 1.1. LOGARITHMIC STRUCTURES 3 Analogously we write N M N or the pullback o the corresponding diagram with reversed arrows, which is the shea o monoids U N (U) M(U) N (U). Let α be a prelog structure on the scheme X. The monoid shea M > α := M α O X := M α α 1 (O X ) O X its naturally into the diagram α 1 (O X ) M α α O X M > α α ι! O X, coming with a unique homomorphism o monoid sheaves M > α O X, denoted α > and given by the local rule m u u α(m). In act, α > : M > α O X is a log structure on X. I α is already a log structure, then α = α >. For the image m 1 o a local section m o M α under M α M α > we will usually just write m Deinition We call α > : M > α O X the log structure associated to the prelog structure α: M α O X. Any morphism o prelog structures ϕ: β α with α a log structure actors through β > uniquely. Associated to each morphism ϕ: β α o prelog structures there is a unique morphism o log structures ϕ > : β > α > called the morphism o log structures associated to ϕ. This deines a unctor ( ) > : prelog X Log X which is let adjoint to the orgetul unctor in the opposite direction Remark The notation M α O X is non-standard; it is used to symbolise that the shea o units O X o the structure shea is a subgroup shea in the enlarged shea M > α. Let : X Y be a morphism o schemes. As usual, we denote the inverse image o a shea G on Y under by 1 G and the direct image o a shea F on X under by F Deinition a) For any (pre)log structure β : M β O Y on Y the inverse image morphism 1 β : 1 M β 1 O Y O X is a prelog structure on X. We denote its associated log structure ( 1 β) > : ( 1 M β ) O X O X by β and write M β := ( 1 M β ) > or its log structure shea. β and M β are called the (logarithmic) pullback o β and M β under, respectively.

16 4 CHAPTER 1. LOGARITHMIC GEOMETRY b) For any log structure α: M α O X on X we have the two morphisms O Y O X and α: M α O X. We deine M α and the morphism α to be the ibred product M α O X O Y and the canonical morphism in the diagram M α α M α α O Y O X, respectively. Since the preimage under o O X is O Y, the morphism α is a log structure. α and M α are called the (logarithmic) direct image o α and M α under, respectively. With regard to characteristic monoid sheaves, or and β as in the deinition, there is a canonical isomorphism 1 M β = M β (c. [19, 1.4.1]) Proposition ([29, III.1.1.5]) Let : X Y be a morphism o schemes. The two unctors : Log Y Log X and : Log X Log Y are adjoint unctors. To be precise, there is a natural isomorphism Remark Hom LogX ( β, α) = Hom LogY (β, α). Since the tensor product o sheaves o monoids involves a sheaiication process, the log structure α > associated to a prelog structure α depends on the chosen topology (Zariski or étale) on X, unless the structure shea M α o α is a shea o unit-integral monoids. This is meant in the ollowing sense: Let υ X : Xét X zar be the canonical continuous map and let α: M α O X be a prelog structure on X zar. Let then α > zar := α > be the associated log structure to α on X zar and α > ét := (υ 1 X α)> the associated log structure to υ 1 X α on X ét. Then we have a natural morphism o prelog structures υ 1 X zar) α (α> > ét (where the second one is a log structure). I the structure shea M α o α is a shea o unit-integral monoids, then this morphism is an isomorphism, i. e. i e: X X is an étale morphism, then the restriction α > ét X zar = υ X (e α > ét ) is equal to the log pullback e α > zar = e α as sheaves on X zar. In particular Γ (e: X X, M > α,ét ) = Γ (X, e M α ) (cp. [29, III.1.1.4]) Charts and Coherence Let X be a scheme and α: M α O X a log structure on X. Let P be a monoid and a: P Γ (X, M α ) a monoid homomorphism. Then a induces trivially a morphism o prelog structures a: α a α, given by a: P X M α, where α a is the prelog structure a α P X M α OX obtained by composition.

17 1.2. LOGARITHMIC SCHEMES Deinition A (global) chart or a log structure α on a scheme X is a monoid homomorphism a: P Γ (X, M α ) such that a > : (α a) > particular, P O X = M α ). α is an isomorphism o log structures (then, in A chart or α at a point x in X is an open/étale neighbourhood U o x together with a chart a: P Γ (U, M). Let P be one o the ollowing properties o a monoid: coherent, domainic, sharp, (unit-/ quasi-)integral, ine, saturated, toric, normal, ree Deinition Let α be a log structure on a scheme X. We say that a chart a: P Γ (X, M α ) o α has the property P i the monoid P has the property P. A log structure α: M α O X on a scheme X is called quasi-coherent i or any point x X there exists a chart a: P Γ (U, M α ) at x. We say that a quasi-coherent log structure α has the property P i or any point x X there exists a chart a: P Γ (U, M α ) at x with that property (an exception to this rule is the case that i or any point x X there exists a ree chart, we call α locally ree). The restriction to an open subscheme o a quasi-coherent log structure is again quasicoherent. Such a restriction preserves any o the above properties P Remark All properties P deined here or log structures, are deined or any shea o monoids [c. Appendix A]. Indeed, a quasi-coherent log structure α is integral (respectively, saturated) i and only i M α,x is integral (saturated) or all points x X. However, a coherent log structure α does almost never have a coherent log structure shea M α in the sense that all o its stalks are coherent. This is due to the act, that the shea o units O X in the ring o regular unctions, which is not aected by the charts, has generally incoherent stalks O X,x at any point x X (compare the remark ollowing deinition II in [29]). 1.2 Logarithmic schemes A logarithmic scheme X will be deined as a scheme with additional structure. Whenever we speak o a shea on X, we mean a shea on the underlying scheme Logarithmic schemes Deinition A logarithmic scheme is a pair X = (X, α X ), where X is a scheme and α X is a log structure on X. Given a log scheme X, we will denote its underlying scheme by X and its log structure by α X : M X O X, where we write M X := M αx and O X := O X, reerring to M X as the logarithmic structure shea o X.

18 6 CHAPTER 1. LOGARITHMIC GEOMETRY A log subscheme o a log scheme X is a subscheme Y X with the log structure j α X induced by its inclusion j : Y X. A morphism o log schemes = (, ): X Y is a morphism o schemes : X Y together with a morphism : α Y α X o log structures on X. A morphism o log schemes : X Y is called (logarithmically) strict i : α Y α X is an isomorphism. It is called logarithmically dominant i is injective, and (logarithmically) semistrict i is surjective. We say that is the underlying morphism o schemes o. From now on, we will usually write,,, etc. or the unctors,,, etc Remark One may also deine to be a morphism o sheaves o monoids such that the diagram M Y M X β α O Y O X commutes. By Deinition and the adjunction property this is equivalent to the deinition given in We will use either orm o according to its useulness. We denote the category o log schemes by LSch. On every scheme S there exists the trivial log structure ι: O S O S and the unctor ( ) ι : Sch LSch, S S ι := (S, ι), is ully aithul. This turns Sch into a ull subcategory o LSch. The inclusion unctor ( ) ι is right adjoint to the unctor orgetting the log structure: Hom LSch (X, S ι ) = Hom Sch (X, S), or X LSch and S Sch (c. [29, III.1.2.1]). I Z is a log scheme, then the category LSch Z is deined as the category which has as objects morphisms o log schemes X Z and as morphisms commutative diagrams o morphisms o log schemes X Y Z, which, by abuse o notation, we will denote by X Y. In the category o log schemes the ibred product o a diagram o log schemes X Y X may be constructed as the ibred product X Y X o the underlying schemes together with the log structure α X α Y α X : M X M Y M X O X Y X deined by α X α Y α X := pr X α X pr Y α Y pr X α X, where we write pr X : X Y X X, pr X : X Y X X and pr Y : X Y X Y or the canonical projections.

19 1.2. LOGARITHMIC SCHEMES Deinition A prelogarithmic ring is a monoid homomorphism a: P A rom a monoid P to the multiplicative monoid o a ring A. A prelog ring homomorphism (π, θ): (a: P A) (b: Q B) is a commutative diagram P θ Q a b A π B, where θ is a monoid homomorphism and π a ring homomorphism. A logarithmic ring is a prelog ring a: P A such that a is logarithmic, i. e. a 1 (A ) = A ; i A is a ield, we speak o a (pre)log ield. A log ring homomorphism is a prelog ring homomorphism between log rings. Given a prelog ring a: P A we denote its associated log ring a > : P > A, setting P > := P A := P a 1 (A ) A. I A is a ring, then we will write A ι or the prelog ring A ι : 1 A Remark Our deinition o a log ring is that o F. Kato (c. [18]). However, A. Ogus uses the term log ring or what we call prelog ring (c. [29, III.1.2.3]). With regard to this does not produce much o a conlict Deinition I a: P A is a (pre)log ring, then Spec a is deined to be the log scheme X which has X := Spec A as its underlying scheme and the log structure α X : P X O X O X associated to the prelog structure a: P X O X. Note that Spec a and Spec a > are equal as log schemes. Hence the at irst sight ambiguous notation Spec A ι, which may be read as Spec(A ι ) or (Spec A) ι, denotes a unique log scheme. More generally, we call a log scheme aine i its underlying scheme is an aine scheme. A logarithmic point is a log scheme X such that X = Spec k or a ield k. The standard prelogarithmic ield o the ield k is the prelog ield κ: N 0 k mapping 0 1 and n 0 i n 1. The standard logarithmic point o the ield k is Spec κ. Note that not every aine log scheme is isomorphic to the spectrum o a (pre)log ring. However, any log scheme with quasi-coherent log structure may be covered by spectra o (pre)log rings Lemma ([29, III.1.5.3]) Let S = Spec k, with k an algebraically closed ield. Since S consists only o one point, any log structure κ on S is given by a monoid homomorphism κ: M k. I M is unit-integral, then the log scheme (S, κ) is (non-canonically) isomorphic to Spec k P or the sharp monoid P := M/k.

20 8 CHAPTER 1. LOGARITHMIC GEOMETRY Deinition Let A be a commutative ring and P a monoid. The P -aine log scheme over A is the log scheme A A [P ] := Spec(P A[P ]) associated to the canonical prelog ring P A[P ]. Its log structure will be denoted α A [P ] and called the canonical log structure on the scheme Spec A[P ]. I θ : Q P is a monoid homomorphism, we write A A [θ]: A A [P ] A A [Q] or simply θ or the morphism o Spec A ι -log schemes given by the ring homomorphism A[θ]: A[Q] A[P ]. This makes A A [ ] a unctor Mon op LSch Spec A ι. In the case A = Z, we simply write A[P ] := A Z [P ] and A[θ] := A Z [θ]. For the trivial monoid P = 1 we get A A [1] = Spec A ι and we have A A [P ] = A A [1] A[1] A[P ] Deinition Let h: Y A[Q] be an A[Q]-scheme and let θ : Q P be a monoid homomorphism. The P -aine log scheme over h is the pullback Y A[Q] A[P ], denoted A h [P ] or A Y/Q [P ]: A h [P ] Y h A[P ] θ A[Q]. We will denote its log structure by α h [P ] and the canonical projections to A[P ] by h[θ] and to Y by A h [θ], or θ h or short. I Q = 1 and i Y = Spec A is aine, then A Y/1 [P ] = A A [P ]. Given a log scheme X and a monoid P, the map Hom LSch (X, A[P ]) Hom Mon (P, Γ (X, M X )), deined by g a g, where a g is the composition P Γ (X, g P ) group isomorphism. We denote its inverse by a g a. g Γ (X, M X ), is a This means, that speciying a global chart a: P Γ (X, M X ) or α X (with P > = M X ) is equivalent to giving a strict morphism o log schemes g : X A[P ]. In act, with the labelling o deinition 1.1.8, a > = g and α > a = g α A[P ] Deinition We say that a log scheme X is logarithmically P (where P is one o the properties as in section 1.1.2) i its log structure α X has the property P. I Q is a property o schemes (respectively, o morphisms o schemes), then we say that a log scheme X (a morphism o log schemes : X Y ) has the property Q i the underlying scheme X (the underlying morphism o schemes ) has that property.

21 1.2. LOGARITHMIC SCHEMES 9 In particular, a log scheme X is logarithmically ine i it can be covered by ine charts. These are local charts a: P Γ (U, M X ) o α X such that P is initely generated and integral (i. e. maps injectively into P grp ). It is called logarithmically saturated i analogously there is a covering by charts with P integral and P = P sat = {x P grp x n P or some n N 0 }. A logarithmically s (or logarithmically normal) log scheme is a log ine and log saturated log scheme. Likewise, a morphism o log schemes is called proper, i its underlying morphism is proper Remark All authors mentioned in the introduction to this chapter use the convention, that a log scheme has property P i its log structure has P. However, this produces conlicts, e. g. when talking about an integral log scheme, where integral can either be read as a property o its underlying scheme or o its log structure. Our convention helps to avoid these conlicts. We denote the ull subcategory o LSch, having as objects the log (quasi-)coherent log schemes, by LSch (q)coh. The ull subcategory o LSch coh o log ine (respectively, log s, log locally ree) log schemes is denoted LSch (LSch s, LSch l ) Deinition For any morphism : X Y o log schemes we deine the shea o monoids M := M αx / α Y = M X / ( M Y ) on X, which we call the relative characteristic shea or the lenience shea o (or o X over Y ) Lemma ([29, III.1.2.8]) I : X Y is a strict morphism o log schemes, then the map : 1 M Y M X induced by is an isomorphism. I it is semi-strict then M = 0. The converses o both statements are true i X is log integral Deinition Let X be a log coherent log scheme. We denote by LLoc X the reduced closed log subscheme supp M X X and call it the logarithmic locus o X or the support o its log structure α X. Its open complement is denoted by X and called the logarithmically trivial locus o X. A point x X lies in X i and only i there is an open/étale neighbourhood U o x such that α X U is trivial. The unctor ( ) : X X rom the category o log schemes with coherent log structure to schemes is right adjoint to the inclusion unctor ( ) ι : Hom LSch coh(s ι, X) = Hom Sch (S, X )

22 10 CHAPTER 1. LOGARITHMIC GEOMETRY Deinition Let X be a log coherent log scheme. The map l X : X N 0, given by l X (x) := rk(m grp X,x), is called the logarithmic rank o X and the number l X (x) the logarithmic rank o X at the point x. The set X ( n) := l 1 X ({0, 1,... n}) is an open subscheme o X or each n N 0. The underlying set o its reduced (closed) complement X ( n+1) is l 1 X (N n+1). We call the locally closed intersection X (n) = X ( n) X ( n) with underlying set l 1 X (n) the logarithmic locus o X o n th order. The scheme structure o each o these subschemes X (n) is, locally at a point x given by the ideal I αx,x O X,x, deined to be the image under α X o the maximal ideal m MX,x o M X,x. This deines a stratiication X = X ( 0) X ( 1)... X ( N) =, N 0, or X. Moreover, codim X X ( n) n or all n (cp. [29, II.2.1.6] and [16, 2.3]). In particular, LLoc X = X ( 1) and X = X (0) Morphisms o log Schemes and Charts As mentioned beore, speciying a global chart a: P Γ (X, M X ) or a log scheme X is equivalent to giving a strict morphism o log schemes g : X A[P ]. A similar statement is true, when talking about charts o morphisms o log structures: Deinition Let : X Y be a morphism o log schemes. A (global) chart o subordinate to a monoid homomorphism θ : Q P is a commutative diagram Q b Γ (Y, M Y ) P θ a Γ (X, M X ) o monoid homomorphisms, where a and b are global charts or α X and α Y respectively. Speciying a chart or subordinate to θ : Q P is equivalent to giving a commutative diagram X g a A[P ] Y g b θ A[Q] o morphisms o log schemes, with g a and g b strict. We will call this the chart diagram o the chart.

23 1.2. LOGARITHMIC SCHEMES 11 This diagram canonically enlarges to X g a/b A b [P ] g b [θ] A[P ] Y θ b g b θ A[Q] with the right square Cartesian, g a = g b [θ] g a/b and all horizontal maps strict; here we write A b [P ] or A gb [P ] and θ b or θ gb. We will reer to it as the extended chart diagram o the chart. Given a morphism o log schemes : X Y, a chart b: Q Γ (Y, M Y ) or α Y and a monoid homomorphism θ : Q P, we have a group isomorphism Hom MonQ (P, Γ (X, M X )) = Hom LSchY (X, A b [P ]), given by a g a/b Proposition ([19, 2.10], [17, 2.14]) Let : X Y be a morphism o log coherent log schemes and let b: Q Γ (Y, M Y ) be a coherent chart or α Y. Then locally on X there exists a chart Q b Γ (Y, M β ) P θ a Γ (X, M α ) comprising b and with P initely generated Deinition Let : X Y be a morphism o log coherent log schemes. We denote by LLoc the reduced closed log subscheme supp M X and call it the lenient locus o. Its open complement is denoted by X () and called the semi-strict locus o (in X). A point x X lies in X () i and only i there is an open/étale neighbourhood Uo x such that the restriction U : U Y is a semi-strict morphism. I is log dominant, then one may replace semi-strict by strict and speak o the strict locus o Deinition Let : X Y be a morphism o log coherent log schemes. We call the map l : X N 0, given by l (x) := rk(m grp,x), the leniency o and the number l (x) the leniency o at x. The set X ( n) () := l 1 ({0, 1,... n}) is an open subscheme o X or each n N 0. The underlying set o its reduced (closed) complement X ( n+1) () is l 1 (N n+1). We call the locally closed intersection X (n) () o X ( n) () and X ( n) () with underlying set l 1 X (n) the lenient locus o o n th order. The scheme structure o each o these subschemes is locally at a point x given by the ideal I MX,x LLoc = X ( 1) () and X () = X (0) (). O X,x (c. [29, II.2.1.6], [16, 2.3]). It is

24 12 CHAPTER 1. LOGARITHMIC GEOMETRY Log derivations and log dierentials Deinition Let : X Y be a morphism o log schemes and E an O X -module. A (logarithmic) derivation o (or o X over Y ) to E is a pair ϑ = (ϑ, Θ), where ϑ Der Y (X, E) and Θ : M X (E, +) is a morphism o sheaves o monoids with the ollowing properties: a) α X (m)θ(m) = ϑ(α X (m)) or any local section m o M X and b) Θ( (n)) = 0 or any local section n o 1 M Y. The set o such derivations is denoted by Der (E). Then the shea Der (E) o (log) derivations o X over Y to E is the shea U Der U (E U ), which is in act an O X -module. I E = O X, we write T or this shea and call it the (logarithmic) tangent shea o or o X over Y Remark Since (E, +) is a shea o groups, the map Θ naturally actors via the morphism o sheaves o groups Θ grp : M grp X (E, +) Deinition and Proposition (cp. [29, IV.1.2.3]) Let : X Y be a morphism o log schemes. There exists an O X -module Ω 1 and a universal derivation (d, dlog) Der (Ω 1 ) such that or any O X-module E the canonical morphism o O X -modules is an isomorphism. Hom OX (Ω 1, E) Der (E), λ (λ d, λ dlog) We write i ( ): T OX Ω 1 O X, ϑ σ i ϑ (σ), or the natural pairing induced by this isomorphism and we call Ω 1 the shea o (logarithmic) dierentials o X over Y Deinition Let Λ X denote the O X -module O X Z M grp, which we call the shea o purely logarithmic dierentials. The shea o log dierentials may be constructed as ] Ω 1 = [Ω 1 Λ X /(K X + K ), X where Ω 1 is the usual shea o Kähler dierentials and where K X is the O X -submodule o Ω 1 Λ X generated by local sections o the orm and K the image o the map (d(α(m)), α(m) m) or local sections m o M X O X Z 1 M grp Y 0 Λ X,

25 1.2. LOGARITHMIC SCHEMES 13 which is the O X -module generated by local sections o the orm (0, 1 (n)) or local sections n o 1 M Y. The universal derivation is then given by d d: O X Ω 1 Ω 1 and dlog: M X O X Z M grp X Ω, m (0, 1 m). Accordingly we will simply write ds or the class o (ds, 0) and dlog m or the class o (0, 1 m). Occasionally, i s = α(m), we write dlog s := dlog m. Given a local section ϑ = (ϑ, Θ) o Der (E) or and E as above, then ϑ is completely determined by Θ. This is due to the act, that O X generates (O X, +) as a shea o Abelian groups, because any local section o O X can locally be written as the sum o at most two sections o O X. Hence, the image o M X under α X generates O X as a shea o Abelian groups. Any local section s o O X can locally be written as s = α(m) or as s = α X (m 1 ) + α X (m 2 ). Then ϑ(s) = sθ(m) or ϑ(s) = α(m 1 )Θ(m 1 ) + α(m 2 )Θ(m 2 ) (cp. [29, IV.1.2.4]). This leads to an alternative construction o Ω 1 as a quotient o Λ X: Proposition ([29, IV & 11]) Let R X Λ X be the subshea o sections, which are locally o the orm α X (m i ) m i i j α X (m j) m j, where m 1,..., m k and m 1,..., m k are local sections o M X such that i α X(m i ) = j α X(m j ) in O X, and let R Λ X be the image o the map O X Z 1 M grp Y Λ X. Then R X and R are O X -submodules o Λ X and there is a unique isomorphism Ω 1 = Λ X /(R X + R ). We will denote the image o a local section m o Λ X in Ω 1 by dlog m. This notation is consistent in the sense that i m is the image o a local section m o M X under M X M grp X, then dlog m, as deined here, is equal to dlog m, as deined above as the image o m under the universal derivation dlog. I : S ι S ι is a morphism o schemes with trivial log structures, then the O X -module o usual Kähler dierentials Ω 1 o is canonically isomorphic to Ω1 by the act that or any local section u o M S ι = O S we have dlog u = u 1 du. Hence, we will identiy both modules in this case, as well as their dual modules T and T. I : X Y is a morphism o log coherent log schemes, then Ω 1 is a quasi-coherent O X- module. I moreover Y is Noetherian and the underlying morphism o schemes : X Y is o inite type, then Ω 1 is coherent (c. [17, 5.1], [29, IV.1.2.9]).

26 14 CHAPTER 1. LOGARITHMIC GEOMETRY Proposition ([29, IV.1.3.1]) For a commutative diagram o morphisms o log schemes X g X Y h Y there is a natural morphism g Ω 1 Ω1 (sending ds to g ds = d(g s) and dlog m to g dlog m := dlog(g m) or local sections s o g 1 O X and m o g 1 M X ), which is an isomorphism i the diagram is Cartesian. I this diagram is Cartesian, then the induced homomorphism Ωh 1 g Ω 1 Ωh 1 is an isomorphism Proposition ([29, IV & 2]) = Ω1 g Let : X Y and g : Y Z be two morphisms o log schemes. Then there is an exact sequence Ωg 1 Ωg 1 Ω 1 0. I we replace the morphism by a strict closed immersion i: X Y deined by a shea o ideals I and i X, Y and Z are log quasi-integral, then there is an exact sequence I/I 2 i Ωg 1 Ωi g 1 0 o O X -modules. We call the O X -module I/I 2 the conormal shea o i. Let : X Y be a morphism o log coherent log schemes. The 1 O Y -linear map d: O X Ω 1 its into a complex Ω with 1 O Y -linear dierentials, called the logarithmic de Rham complex o : Proposition ([29, V.2.1.1]) Let : X Y be a morphism o log coherent log schemes. There exists a complex o O X -modules Ω with 1 O Y -linear dierentials d i : Ω i Ωi+1 a) Ω i = i Ω 1 ; b) d 0 = d: O X Ω 1 ; c) d 1 (dlog m) = 0 or each local section m o M X ; such that d) d i+j (σ σ ) = (d i σ) σ + ( 1) i σ (d j σ ) or local sections σ o Ω i and σ o Ω j.

27 1.2. LOGARITHMIC SCHEMES Logarithmic ininitesimal thickenings Deinition A logarithmic (ininitesimal) thickening is a strict closed immersion i: T T o log schemes such that the ideal shea I o T in T is nilpotent and such that the subgroup 1 + I O T operates reely on M T. We say that the log thickening has (at most) order n i I n+1 = 0. I T is log quasi-integral, then the action o 1 + I on M T is automatically ree. For the Zariski topology, in a log thickening the underlying topological spaces o T and T are homeomorphic and are thus identiied. For the étale topology the same is true or log thickenings o inite order (c. [29, IV.2.1.4]) Proposition ([29, IV & 3]) Let i: T T be a log thickening with ideal I. Then a) The diagram O T M T is Cartesian and cocartesian; i i O T i i M T b) Ker(O T i O T ) = Ker(Mgrp T i M grp T ) = 1 + I (observe, that i O T = i O T when regarding O T as the structure shea o the trivial log structure on T ); c) The diagram M T M grp T is Cartesian; M T M grp T d) T is log coherent, log integral, log ine, log saturated or log ree i and only i T is. I a: P Γ (T, M T ) is a chart or T, then i a: P Γ (T, M T ) is a chart or T and the converse is true i T is log quasi-integral Logarithmic smoothness Formal smoothness or schemes was deined by A. Grothendieck in [12, III.1] via the ininitesimal liting property. A morphism o schemes is smooth i and only i it has the ininitesimal liting property and is locally o inite presentation. In analogy one deines (ormal) log smoothness.

28 16 CHAPTER 1. LOGARITHMIC GEOMETRY Deinition ([19, 3.3], [29, IV.3.1.1]) A morphism : X Y o log schemes is ormally smooth (respectively, ormally unramiied, ormally étale) i or every n and every n-th order log thickening i: T T with a commutative diagram T i ϕ X T ϕ Y there exists at least one (at most one, exactly one) morphism ϕ: T X liting ϕ in this diagram. A ormally smooth (respectively, ormally étale) morphism : X Y is called logarithmically smooth (logarithmically étale) i X and Y are log coherent and its underlying morphism : X Y is locally o inite presentation. The usual statements about composition and base change o smooth and étale morphisms hold or log smooth and log étale morphisms. I is log smooth, then Ω 1 is locally ree (c. [29, IV.3.2.1]). A morphism o log coherent log schemes is ormally unramiied i and only i Ω 1 = 0 (c. [29, IV.3.1.3]) Proposition ([29, IV & 4]) Let : X Y and g : Y Z be two morphisms o o log coherent log schemes. a) I is log smooth, then the sequence 0 Ωg 1 Ωg 1 Ω1 splits. b) I g is log smooth and the sequence 0 Ωg 1 Ωg 1 Ω1 then is log smooth. 0 is exact and 0 is split exact, c) I is log étale, then Ωg 1 Ωg 1 is an isomorphism Proposition ([29, IV.3.1.6]) Let : X Y be a strict morphism o log coherent log schemes. I the underlying morphism o schemes is ormally smooth (respectively, ormally étale, ormally unramiied) then the same is true o. The converse holds i Y is log unit-integral. This leads to the ollowing deinition: Deinition Let : X Y be a morphism o log coherent log schemes. We say that is smooth (respectively, étale, unramiied) i is strict and log smooth (log étale, ormally unramiied) Proposition ([29, IV.3.1.8]) Let θ : Q P be a homomorphism o initely generated monoids and let : A A [P ] A A [Q] be the corresponding aine morphism over a commutative ring A. Then is log smooth (respectively, log étale) i and only i Ker(θ grp ) and the torsion part o Cok(θ grp ) (Ker(θ grp ) and Cok(θ grp )) are inite groups the order o which is invertible in A.