TY - JOUR AB - We give a simple argument to prove Nagai’s conjecture for type II degenerations of compact hyperkähler manifolds and cohomology classes of middle degree. Under an additional assumption, the techniques yield the conjecture in arbitrary degree. This would complete the proof of Nagai’s conjecture in general, as it was proved already for type I degenerations by Kollár, Laza, Saccà, and Voisin [10] and independently by Soldatenkov [18], while it is immediate for type III degenerations. Our arguments are close in spirit to a recent paper by Harder [8] proving similar results for the restrictive class of good degenerations. AU - Huybrechts, D. AU - Mauri, Mirko ID - 13268 IS - 1 JF - Mathematical Research Letters SN - 1073-2780 TI - On type II degenerations of hyperkähler manifolds VL - 30 ER - TY - JOUR AB - In this paper, we determine the motivic class — in particular, the weight polynomial and conjecturally the Poincaré polynomial — of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank bundle on P1. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss–Leclerc–Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the four-dimensional cases are expected to be of type ALF. AU - Hausel, Tamás AU - Wong, Michael Lennox AU - Wyss, Dimitri ID - 14244 IS - 4 JF - Proceedings of the London Mathematical Society SN - 0024-6115 TI - Arithmetic and metric aspects of open de Rham spaces VL - 127 ER - TY - CHAP AB - We construct for each choice of a quiver Q, a cohomology theory A, and a poset P a “loop Grassmannian” GP(Q,A). This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a “dilation” torus D⊆G2m gives a quantization GPD(Q,A). This construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program (Mirković, Some extensions of the notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić issue. No. 532, 53–74, 2017) and on the construction of affine quantum groups from generalized cohomology theories (Yang and Zhao, Quiver varieties and elliptic quantum groups, preprint. arxiv1708.01418). AU - Mirković, Ivan AU - Yang, Yaping AU - Zhao, Gufang ED - Baranovskky, Vladimir ED - Guay, Nicolas ED - Schedler, Travis ID - 12303 SN - 2297-0215 T2 - Representation Theory and Algebraic Geometry TI - Loop Grassmannians of Quivers and Affine Quantum Groups ER - TY - JOUR AB - For a Seifert fibered homology sphere X we show that the q-series invariant Zˆ0(X; q) introduced by Gukov-Pei-Putrov-Vafa, is a resummation of the Ohtsuki series Z0(X). We show that for every even k ∈ N there exists a full asymptotic expansion of Zˆ0(X; q) for q tending to e 2πi/k, and in particular that the limit Zˆ0(X; e 2πi/k) exists and is equal to the WRT quantum invariant τk(X). We show that the poles of the Borel transform of Z0(X) coincide with the classical complex Chern-Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, C)-connections. AU - Mistegaard, William AU - Andersen, Jørgen Ellegaard ID - 9977 IS - 2 JF - Journal of the London Mathematical Society TI - Resurgence analysis of quantum invariants of Seifert fibered homology spheres VL - 105 ER - TY - JOUR AB - We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of C∗-actions on semiprojective varieties, C∗ characters of indices of C∗-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier–Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs bundles. AU - Hausel, Tamás AU - Hitchin, Nigel ID - 10704 JF - Inventiones Mathematicae SN - 0020-9910 TI - Very stable Higgs bundles, equivariant multiplicity and mirror symmetry VL - 228 ER -