Fundamental and divisor class group: finiteness and interplay
Final Report Abstract
This project in the Walter-Benjamin-Program aimed to investigate (the interplay of) two important invariants of varieties with klt-singularities, namely: 1. The fundamental group (of the smooth locus). 2. The divisor class group and the Cox ring. In order to do so, the first goal - which has partly been achieved already in preparatory work - was to define Cox rings (which till then only have been defined for varieties) for local rings of klt singularities. The findings have been published joint with Joaquın Moraga. Not only we defined Cox rings for different local models as e.g. the Zariski local ring and its Henselization as well as its completion with respect to their respective divisor class groups, but also we introduced the iteration of Cox rings in this local setting and proved its finiteness. The interplay of the fundamental group (finite by the PI’s work and the iteration of Cox rings leads to the existence of a simply connected factorial canonical (scfc) cover, proven as well. We carried this investigation from local to global joint with Joaquın Moraga as well. In this context, the group of Weil-modulo-Cartier divisors takes the place of the local class group. We prove this group to be finite as well for projective klt varieties. However, as for the case of tale fundamental groups, not all good properties hold in the global case: the existence of the scfc cover does not, not even for varieties with toric singularities. An unforeseen major application of these investigations is the kltness of reductive quotients, published together with Daniel Greb, Kevin Langlois, and Joaquın Moraga. This generalizes Boutots famous work about rationality of reductive quotients. Applications include the kltness of Artin stacks, moduli spaces of K-polystable Fano manifolds, collapsings of a homogeneous bundles, certain symplectic quotients of Kähler manifolds, and the fact that reductive quotients of Fano type varieties are of Fano type. In the second part of the project, the investigation was focused on the fundamental groups of klt varieties, in particular with trivial or nef anticanonical divisor. On the one hand, this made a necessary the investigation of orbifold and degenerate Kähler metrics, which is one approach coming from the manifold case to investigate the fundamental group. First findings in this direction have been published. On the other hand, we were able to prove first results in low dimensions (≤ 3) with the known methods, together with Fernando Figueroa. In this work, we also investigate the relation of the fundamental group and a certain invariant, the coregularity, relying precisely on these low-dimensional findings.
Publications
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ODD Metrics.
L. Braun
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The Jordan property for local fundamental groups. Geometry & Topology, 26(1), 283-319.
Braun, Lukas; Filipazzi, Stefano; Moraga, Joaquín & Svaldi, Roberto
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Komplexe Analysis – Differential and Algebraic methods in Kähler spaces. Oberwolfach Reports, 20(2), 965-1030.
Eyssidieux, Philippe; Hwang, Jun-Muk; Kebekus, Stefan & Păun, Mihai
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Orbifold Kähler–Einstein metrics on projective toric varieties. Bulletin of the London Mathematical Society, 55(6), 2743-2748.
Braun, Lukas
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Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs.
L. Braun & F. Figueroa
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Iteration of Cox rings of klt singularities. Journal of Topology, 17(1).
Braun, Lukas & Moraga, Joaquín
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Reductive covers of klt varieties. Revista Matemática Iberoamericana, 40(5), 1701-1730.
Braun, Lukas & Moraga, Joaquín
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Reductive quotients of klt singularities. Inventiones mathematicae, 237(3), 1643-1682.
Braun, Lukas; Greb, Daniel; Langlois, Kevin & Moraga, Joaquín
