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22
A DEGENERATION FORMULA OF GWINVARIANTS
, 2001
"... This is the sequel to the paper [Li]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [Li]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth var ..."
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Cited by 49 (2 self)
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This is the sequel to the paper [Li]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [Li]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety. Based on these, we prove a degeneration formula of the GromovWitten invariants.
Toric degenerations of toric varieties and tropical curves
 Duke Math. J
"... Abstract. We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory. Conte ..."
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Cited by 29 (3 self)
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Abstract. We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory. Contents
The StromingerYauZaslow conjecture: From torus fibrations to degenerations
 IN PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
, 2008
"... ..."
Log Smooth Deformation and Moduli of Log Smooth Curves
, 1996
"... The aim of this paper is to define a reasonable moduli theory of log smooth curves which recovers the classical DeligneKnudsenMumford moduli of pointed stable curves. Introduction Moduli theory in the framework of log geometry aims to construct good and natural compactifications of moduli of a ..."
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Cited by 9 (1 self)
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The aim of this paper is to define a reasonable moduli theory of log smooth curves which recovers the classical DeligneKnudsenMumford moduli of pointed stable curves. Introduction Moduli theory in the framework of log geometry aims to construct good and natural compactifications of moduli of algebraic varieties. The motivating philosophy is that, since log smoothness includes some degenerating objects like semistable reductions, etc., the moduli space of log smooth objects should be already compactified, once its existence has been established. In this paper we will develop the theory of moduli of log smooth curves, which is expected to give a compactification of the classical moduli of algebraic curves. This theory should be a starting point and give a prototype of the future study of log moduli theory. Let us give a brief summary: In the next section we will define socalled log curves, which seems the most natural counterpart of smooth curves. Then, in Theorem 1.3, we will se...
Relative log Poincaré lemma and relative log de Rham theory
 Duke Math. J
, 1998
"... In this paper we will generalize the classical relative Poincar'e lemma in the framework of log geometry. Like the classical Poincar'e lemma directly implies the de Rham theorem, the comparison between de Rham and Betti cohomologies, our log Poincar'e lemma yields the formula which gives integral st ..."
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Cited by 6 (1 self)
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In this paper we will generalize the classical relative Poincar'e lemma in the framework of log geometry. Like the classical Poincar'e lemma directly implies the de Rham theorem, the comparison between de Rham and Betti cohomologies, our log Poincar'e lemma yields the formula which gives integral structures of hyperdirect images of the log de Rham complexes; these integral structures are nothing but the integral structures of degenerate VMHS in the semistable degeneration case. We will also develop the relative log de Rham theory for semistable degeneration and recover the wellknown result of Steenbrink. 1 Introduction 1.1 De RhamHodge theory and log geometry Let f : X ! Y be a smooth morphism of complex manifolds, and\Omega ffl X=Y the relative de Rham complex. Then the famous classical Poincar'e lemma asserts that the natural morphism f 01 O Y 0!\Omega ffl X=Y of complexes is a quasiisomorphism. The Poincar'e lemma implies, providing f is proper, the comparison of the...
Compact moduli of hyperplane arrangements
, 2008
"... The minimal model program suggests a compactification of the moduli space of hyperplane arrangements which is a moduli space of stable pairs. Here, a stable pair consists of a scheme X which is a degeneration of projective space and a divisor D = D1 + · · · + Dn on X which is a limit of hyperplane ..."
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Cited by 5 (0 self)
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The minimal model program suggests a compactification of the moduli space of hyperplane arrangements which is a moduli space of stable pairs. Here, a stable pair consists of a scheme X which is a degeneration of projective space and a divisor D = D1 + · · · + Dn on X which is a limit of hyperplane arrangements. For example, in the 1dimensional case, the stable pairs are stable curves of genus 0 with n marked points. Kapranov has defined an alternative compactification using his Chow quotient construction, which may be described fairly explicitly. We prove that these two compactifications coincide. We deduce a description of all stable pairs.
UNIVERSAL LOG STRUCTURES ON SEMISTABLE VARIETIES
"... Abstract. Fix a morphism of schemes f: X → S which is flat, proper, and “fiberbyfiber semistable”. Let IV LS be the functor on the category of log schemes over S which to any T associates the isois a log structure on morphism classes of pairs (MX, f b), where MX X ×S T and f b: pr ∗ 2MT → MX is ..."
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Cited by 4 (1 self)
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Abstract. Fix a morphism of schemes f: X → S which is flat, proper, and “fiberbyfiber semistable”. Let IV LS be the functor on the category of log schemes over S which to any T associates the isois a log structure on morphism classes of pairs (MX, f b), where MX X ×S T and f b: pr ∗ 2MT → MX is a morphism of log structures making (X ×S T, MX) → T a log smooth, integral, and vertical morphism. The main result of this paper is that IV LS is representable by a log scheme. In the course of the proof we also generalize results of F. Kato on the existence of log structures of embedding and semistable type. 1.
Inseparable local uniformization
"... The aim of this paper is to prove that an algebraic variety over a field can be desingularized locally along a valuation after a purely inseparable alteration. Zariski was first to study the problem of desingularizing algebraic varieties along valuations. He called this problem local uniformization ..."
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Cited by 4 (0 self)
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The aim of this paper is to prove that an algebraic variety over a field can be desingularized locally along a valuation after a purely inseparable alteration. Zariski was first to study the problem of desingularizing algebraic varieties along valuations. He called this problem local uniformization of valuations and observed that it should
Mochizuki’s crysstable bundles: A lexicon and applications
 Publications of RIMS, to appear. BRIAN OSSERMAN
"... Abstract. Mochizuki’s work on torally crysstable bundles [9] has extensive implications for the theory of logarithmic connections on vector bundles of rank 2 on curves, once the language is translated appropriately. We describe how to carry out this translation, and give two classes of applications ..."
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Cited by 4 (2 self)
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Abstract. Mochizuki’s work on torally crysstable bundles [9] has extensive implications for the theory of logarithmic connections on vector bundles of rank 2 on curves, once the language is translated appropriately. We describe how to carry out this translation, and give two classes of applications: first, one can conclude almost immediately certain results classifying Frobeniusunstable vector bundles on curves; and second, it follows from the results of [13] that one also obtains results on rational functions with prescribed ramification in positive characteristic. 1.