Two ways to degenerate the Jacobian are the same

A basic technique for studying a family of Jacobian varieties is to extend the family by adding degenerate fibers. Constructing an extension requires a choice of fibers, and one typically chooses to include either degenerate group varieties or degenerate moduli spaces of sheaves. Here we relate these two different approaches when the base of the family is a regular, 1-dimensional scheme such as a smooth curve. Specifically, we provide sufficient conditions for the line bundle locus in a family of compact moduli spaces of pure sheaves to be isomorphic to the N\'eron model. The result applies to moduli spaces constructed by Eduardo Esteves and Carlos Simpson, extending results of Busonero, Caporaso, Melo, Oda, Seshadri, and Viviani.Comment: Preprint updated to match published version. Previously appeared as "Degenerating the Jacobian: the N\'eron Model versus Stable Sheaves

Norm estimates and asymptotic faithfulness of the quantum SU(n)SU(n) representations of the mapping class groups

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operators of compact K\"ahler manifolds and in [10] using skein theory.Comment: Geometriae Dedicata (online), 10 pages, minor change

Vector bundles on proper toric 3-folds and certain other schemes

We show that a proper algebraic n-dimensional scheme Y admits nontrivial vector bundles of rank n, even if Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional locus in only finitely many points. Moreover, there are such vector bundles with arbitrarily large top Chern number. Applying this to toric varieties, we infer that every proper toric threefold admits such vector bundles of rank three. Furthermore, we describe a class of higher-dimensional toric varieties for which the result applies, in terms of convexity properties around rays.Comment: 28 pages, minor changes, to appear in Trans. Amer. Math. So

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Assessing and refining mappings to RDF to improve dataset quality

RDF dataset quality assessment is currently performed primarily after data is published. However, there is neither a systematic way to incorporate its results into the dataset nor the assessment into the publishing workflow. Adjustments are manually -but rarely- applied. Nevertheless, the root of the violations which often derive from the mappings that specify how the RDF dataset will be generated, is not identified. We suggest an incremental, iterative and uniform validation workflow for RDF datasets stemming originally from (semi-) structured data (e.g., CSV, XML, JSON). In this work, we focus on assessing and improving their mappings. We incorporate (i) a test-driven approach for assessing the mappings instead of the RDF dataset itself, as mappings reflect how the dataset will be formed when generated; and (ii) perform semi-automatic mapping refinements based on the results of the quality assessment. The proposed workflow is applied to diverse cases, e.g., large, crowdsourced datasets such as DBpedia, or newly generated, such as iLastic. Our evaluation indicates the efficiency of our workflow, as it significantly improves the overall quality of an RDF dataset in the observed cases

A curvature formula associated to a family of pseudoconvex domains

We shall give a definition of the curvature operator for a family of weighted Bergman spaces $\{\mathcal H_t\}$ associated to a smooth family of smoothly bounded strongly pseudoconvex domains $\{D_t\}$. In order to study the boundary term in the curvature operator, we shall introduce the notion of geodesic curvature for the associated family of boundaries $\{\partial D_t\}$. As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for $\{\mathcal H_t\}$ and its applications to triviality of fibrations are also given in this paper.Comment: 35 pages, to appear in Annales de l'Institut Fourie