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  1. ; ; ; (, Compositio Mathematica)
    null (Ed.)
    We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $$X \longrightarrow B$$ with singular fibre over $$b_0\in B$$ yields a family $$\mathscr {M}(X/B,\beta ) \longrightarrow B$$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $$b_0$$ in terms of rigid tropical maps to the tropicalization of $X/B$ . This generalizes one aspect of known results in the case that the fibre $$X_{b_0}$$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples. 
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  2. ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract In this paper, we study {\mathbb{A}^{1}} -connected varieties from log geometry point of view, and prove a criterion for {\mathbb{A}^{1}} -connectedness. As applications, we provide many interesting examples of {\mathbb{A}^{1}} -connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne conjecture characterizing projective spaces and affine spaces. 
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  3. ; (, Journal of Algebraic Geometry)
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  4. ; ; ; (, Journal of the European Mathematical Society)
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  5. ; (, Mathematische Annalen)
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  6. ; (, International Mathematics Research Notices)
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  7. ; ; ; ; (, Nonarchimedean and Tropical Geometry)
    We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities. 
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