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 Gromov-Witten invariants.

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 algebraic-geometric and relies on degeneration techniques and log deformation theory. Contents

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The aim of this paper is to define a reasonable moduli theory of log smooth curves which recovers the classical Deligne-Knudsen-Mumford 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 so-called log curves, which seems the most natural counterpart of smooth curves. Then, in Theorem 1.3, we will se...

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 well-known result of Steenbrink. 1 Introduction 1.1 De Rham-Hodge 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 quasi-isomorphism. The Poincar'e lemma implies, providing f is proper, the comparison of the...

Abstract. Fix a morphism of schemes f: X → S which is flat, proper, and “fiber-by-fiber semi-stable”. Let IV LS be the functor on the category of log schemes over S which to any T associates the iso-is 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 semi-stable type. 1.

Abstract. We construct algebraic moduli stacks of log structures and give stack-theoretic interpretations of K. Kato’s notions of log flat, log smooth, and log étale morphisms. In the last section we describe the local structure of these moduli stacks in terms of toric stacks. 1.

Introduction. Let f: X → B be a smooth separated morphism of noetherian algebraic schemes of relative dimension N. Let n be a positive integer invertible on B and we write µn,X (resp. µn,B) for the étale sheaf of nth roots of unity on X (resp. on B). Then, as is well-known, we have a canonical relative trace morphism trf: Rf!µ

Let C → S be a proper flat family of stable curves, smooth over S0 = S � ∆. One can associate a flat family f: SU C(r, d) → S0 of moduli spaces SU Cs (r, d) of semistable vector bundles of rank r and degree d with fixed determinant over the