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Authors:Heuer; Ben Pages: 1433 - 1466 Abstract: For any smooth proper rigid space over a complete algebraically closed extension of we give a geometrisation of the -adic Simpson correspondence of rank one in terms of analytic moduli spaces: the -adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of -line bundles. As an application, we study a major open question in -adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the -adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters in terms of moduli spaces: for projective over , it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group PubDate: 2024-05-21 DOI: 10.1112/S0010437X24007024
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Authors:Kotelskiy; Artem, Watson, Liam, Zibrowius, Claudius Pages: 1467 - 1524 Abstract: When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal -grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant and the Khovanov invariant that were developed by the authors in previous works. PubDate: 2024-05-20 DOI: 10.1112/S0010437X24007152
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Authors:Carocci; Francesca, Orecchia, Giulio, Wyss, Dimitri Pages: 1525 - 1550 Abstract: We define -adic or invariants for moduli spaces of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field . Our definition relies on a canonical measure on the -analytic manifold associated to and the invariants are integrals of natural gerbes with respect to . A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a -independence result for these invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement o... PubDate: 2024-05-30 DOI: 10.1112/S0010437X24007176
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Authors:Krylov; Igor, Okada, Takuzo, Paemurru, Erik, Park, Jihun Pages: 1551 - 1595 Abstract: The -inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type , and obtain a -inequality for points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree over satisfying the -condition, all of which have at most terminal singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a point which is not an ordinary double point. PubDate: 2024-05-29 DOI: 10.1112/S0010437X24007164
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over a complete algebraically closed extension 
of 
we give a geometrisation of the 
-adic Simpson correspondence of rank one in terms of analytic moduli spaces: the 
-adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of 
-line bundles. As an application, we study a major open question in 
-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the 
-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters 
in terms of moduli spaces: for projective 
over 
, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group 
-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant 
and the Khovanov invariant 
that were developed by the authors in previous works.
-adic 
or 
invariants for moduli spaces 
of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field 
. Our definition relies on a canonical measure 
on the 
-analytic manifold associated to 
and the 
invariants are integrals of natural 
gerbes with respect to 
. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a 
-independence result for these 
invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement o...
-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type 
, and obtain a 
-inequality for 
points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree 
over 
satisfying the 
-condition, all of which have at most terminal 
singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a 
point which is not an ordinary double point.
