Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures

Steenbrink, J. H. M.

doi:10.1007/bf01446621

Cited by 45 publications

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“…, ζ (λ) n )} and a transition system {(u (λµ) i , σ (λµ) )} such that (11) holds. Then, each X λ has a log structure by (12). Moreover, there exists a system of isomorphisms {φ λµ } defined by (13).…”

Section: Example 2: Smoothings Of Normal Crossing Varietiesmentioning

confidence: 99%

Log smooth deformation theory

Kato¹

1996

Tohoku Math. J. (2)

138

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This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa.

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Section: Example 2: Smoothings Of Normal Crossing Varietiesmentioning

confidence: 99%

Log smooth deformation theory

Kato¹

1996

Tohoku Math. J. (2)

138

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“…The Lemmas 2.3 and 2.4 reduce the proof to the case when X = X ′ . In this case our assertion is proven in ( [KawNam], p. 405-406 and [St3], §5.6).…”

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confidence: 62%

Motivic integral of K3 surfaces over a non-archimedean field

Stewart

Vologodsky

2011

Advances in Mathematics

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We prove a formula expressing the motivic integral ([LS]) of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of Abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces. Preliminaries.2.1. Clemens Polytope and nerve of a strictly semi-stable scheme. Let R be a complete discrete valuation ring with residue field k and fraction field K. Recall that a scheme X of finite type over spec R is strictly semi-stable if every point x ∈ X has a Zariski neighborhood x ∈ U ⊂ X such that the morphism U → spec R factors through anétale morphismfor a uniformizer t of K. If k is perfect, X is a strictly semi-stable scheme if and only if it is regular and flat over R, the generic fiber X = X × R K is smooth over K and the special fiber Y = X × R k is a reduced strictly normal crossing divisor on X.2 There is an extensive literature on maximally degenerate Calabi-Yau varieties over C((t)). See e.g.[Mo1], [LTY].

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“…Moreover it is trivial that this sheaf defines a variation of Hodge structure over SnT. The main theorem in this article, Theorem (6.10), states that for every boundary point s on T there exists, under a certain Kähler condition, a Q-mixed Hodge structure H Q ; W ; F such that the C-vector space H C is isomorphic to the C-vector space R q f X=S log D Cs (where Cs denotes the residue field at the point s) and, via this isomorphism, the filtration F on H C is identified with the filtration on R q f X=S log D Cs obtained from the stupid filtration on X=S log D. In Section 4 we construct a cohomological mixed Hodge complex which gives us such a candidate of the 'limiting' mixed Hodge structure by an elementary way following the idea of Steenbrink in [12], [13] and of Tu in [15]. Here we should mention L.-H. Tu's Ph.D. thesis [15], in which he studied the same problem and proposed a way of constructing such a cohomological complex.…”

Section: Introductionmentioning

confidence: 99%

“…Our cohomological mixed Hodge complex constructed in Section 4 is the same as L.-H. Tu's at the C-structure level. But the construction of the underlying Q-structure is different from his but deeply influenced by the work of Steenbrink in [13]. In Section 1 we present some basic facts which we need later.…”

Section: Introductionmentioning

confidence: 99%

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Fujisawa

1999

Compositio Mathematica

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Abstract. We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink's results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge-De Rham spectral sequence for this case.Mathematics Subject Classifications (1991): 14C30, 14D07, 32G20.

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Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures

Cited by 45 publications

References 8 publications

Log smooth deformation theory

Log smooth deformation theory

Motivic integral of K3 surfaces over a non-archimedean field

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