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1. The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities

   https://doi.org/10.46298/epiga.2025.12186  

   Tighe, Benjamin (March 2025, Épijournal de Géométrie Algébrique)

   We extend results of Looijenga--Lunts and Verbitsky and show that the total Lie algebra $$\mathfrak g$$ for the intersection cohomology of a primitive symplectic variety $$X$$ with isolated singularities is isomorphic to $$\mathfrak g \cong \mathfrak{so}\left(\left(IH^2(X, \mathbb Q), Q_X\right)\oplus \mathfrak h\right),$$ where $$Q_X$$ is the intersection Beauville--Bogomolov--Fujiki form and $$\mathfrak h$$ is a hyperbolic plane. This gives a new, algebraic proof for irreducible holomorphic symplectic manifolds which does not rely on the hyperk\"ahler metric. Along the way, we study the structure of $$IH^*(X, \mathbb Q)$$ as a $$\mathfrak{g}$$-representation -- with particular emphasis on the Verbitsky component, multidimensional Kuga--Satake constructions, and Mumford--Tate algebras -- and give some immediate applications concerning the $P = W$ conjecture for primitive symplectic varieties. Comment: 41 pages; Final journal version; new subsection on LLV algebra for symplectic orbifolds 

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2. Prescribing $$Q$$-curvature on even-dimensional manifolds with conical singularities

   https://doi.org/10.4171/RMI/1543  

   Jevnikar, Aleks; Sire, Yannick; Yang, Wen (February 2025, Revista Matemática Iberoamericana)

   On a2m-dimensional closed manifold, we investigate the existence of prescribedQ-curvature metrics with conical singularities. We present here a general existence and multiplicity result in the supercritical regime. To this end, we first carry out a blow-up analysis of a2mth-order PDE associated to the problem, and then apply a variational argument of min-max type. Form>1, this seems to be the first existence result for supercritical conic manifolds different from the sphere. 

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3. Lattice cohomology and q -series invariants of 3-manifolds

   https://doi.org/10.1515/crelle-2022-0096  

   Akhmechet, Rostislav; Johnson, Peter K.; Krushkal, Vyacheslav (January 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q -series which satisfy some remarkable modularity properties and recover SU ⁢ ( 2 ) {{\rm SU}(2)} quantum invariants of 3-manifolds at roots of unity.In particular, our work gives rise to a 2-variable refinement of the Z ^ {\widehat{Z}} -invariant. 

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4. Spherical conical metrics and harmonic maps to spheres

   https://doi.org/10.1090/tran/8578  

   Karpukhin, Mikhail; Zhu, Xuwen (February 2022, Transactions of the American Mathematical Society)

   A spherical conical metric g g on a surface Σ \Sigma is a metric of constant curvature 1 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2 π 2\pi . The eigenfunctions of the Friedrichs Laplacian Δ g \Delta _g with eigenvalue λ = 2 \lambda =2 play a special role in this problem, as they represent local obstructions to deformations of the metric g g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2 2 -eigenfunctions, given in terms of a certain meromorphic data on Σ \Sigma . As an application we give a description of all 2 2 -eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2 2 -eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2 2 -eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2 2 -eigenfunctions. 

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5. Lagrangian skeleta and plane curve singularities

   https://doi.org/10.1007/s11784-022-00939-8  

   Casals, Roger (June 2022, Journal of Fixed Point Theory and Applications)

   Abstract We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs $$(\mathbb {C}^2,\Lambda )$$ ( C 2 , Λ ) and Weinstein 4-manifolds $$W(\Lambda )$$ W ( Λ ) associated to max-tb Legendrian representatives of algebraic links $$\Lambda \subseteq (\mathbb {S}^3,\xi _\text {st})$$ Λ ⊆ ( S 3 , ξ st ) . We provide computations of Legendrian and Weinstein invariants, and discuss the contact topological nature of the Fomin–Pylyavskyy–Shustin–Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems. 

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