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1. Normal Crossings Singularities for Symplectic Topology: Structures

   https://doi.org/10.1007/s10114-024-2042-4  

   Farajzadeh-Tehrani, Mohammad; Mclean, Mark; Zinger, Aleksey (January 2024, Acta Mathematica Sinica, English Series)

   Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures have applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi’s formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor. 

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2. Logarithmic resolution via multi-weighted blow-ups

   https://doi.org/10.46298/epiga.2024.9793  

   Abramovich, Dan; Quek, Ming Hao (October 2024, Épijournal de Géométrie Algébrique)

   We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka. Specifically, for a singular, reduced closed subscheme $$X$$ of a smooth scheme $$Y$$ over a field of characteristic zero, we resolve the singularities of $$X$$ by taking proper transforms $$X_i \subset Y_i$$ along a sequence of multi-weighted blow-ups $$Y_N \to Y_{N-1} \to \dotsb \to Y_0 = Y$$ which satisfies the following properties: (i) the $$Y_i$$ are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of $$X_i$$, and witness on $$X_{i+1}$$ an immediate improvement in singularities; (iii) and finally, the singular locus of $$X$$ is transformed into a simple normal crossing divisor on $$X_N$$. Comment: Final published version 

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3. On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

   https://doi.org/10.4171/JEMS/1485  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (July 2024, Journal of the European Mathematical Society)

   We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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4. Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan–Tian Perturbations

   https://doi.org/10.1007/s42543-023-00069-1  

   Farajzadeh-Tehrani, Mohammad (May 2023, Peking Mathematical Journal)

   In a previous paper (Farajzadeh-Tehrani in Geom Topol 26:989–1075, 2022), for any logarithmic symplectic pair (X, D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural Fredholm setup for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan–Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove that the subspace of simple maps in each stratum is cut transversely. Such perturbations enable a geometric construction of Gromov–Witten type invariants for certain semi-positive pairs (X, D) in arbitrary genera. In future works, we will use local perturbations and a gluing theorem to construct log Gromov–Witten invariants of arbitrary such pair (X, D). 

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5. Algebraic approximation and the decomposition theorem for Kähler Calabi–Yau varieties

   https://doi.org/10.1007/s00222-022-01096-y  

   Bakker, Benjamin; Guenancia, Henri; Lehn, Christian (June 2022, Inventiones mathematicae)

   Abstract We extend the decomposition theorem for numerically K -trivial varieties with log terminal singularities to the Kähler setting. Along the way we prove that all such varieties admit a strong locally trivial algebraic approximation, thus completing the numerically K -trivial case of a conjecture of Campana and Peternell. 

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