1,721,215 research outputs found
Crepant resolutions of stratified varieties via gluing
Full text linkLet be a variety with a stratification into smooth locally closed subvarieties such that is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset the set of isomorphism classes of locally projective crepant resolutions of defines an -constructible sheaf of sets. Thus, for each stratum and basepoint , the fundamental group acts on the set of germs of projective crepant resolutions at , leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely.
We give criteria for a locally projective crepant resolution to be globally projective. We show that the sheafification of the presheaf of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity . We show the gerbes are automatically trivial if is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative and Nakajima quiver varieties, as well as to canonical threefold singularities.33 pages, comments welcome, fixed issue with proof that locally projective resolutions glue to a globally projective resolution, added subsection 3.3 to further explore the global projectivity issu
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Exceptional Collections for Mirrors of Invertible Polynomials
Full text linkWe prove the existence of a full exceptional collection for the derived
category of equivariant matrix factorizations of an invertible polynomial with
its maximal symmetry group. This proves a conjecture of Hirano--Ouchi. In the
Gorenstein case, we also prove a stronger version of this conjecture due to
Takahashi. Namely, that the full exceptional collection is strong.Comment: 15 pages, minor revision, to appear in Math
Multiplicative preprojective algebras are 2-Calabi-Yau
Full text linkWe prove that multiplicative preprojective algebras, defined by
Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers
containing unoriented cycles. If the quiver is not itself a cycle, we show that
the center is trivial, and hence the Calabi-Yau structure is unique. If the
quiver is a cycle, we show that the algebra is a non-commutative crepant
resolution of its center, the ring of functions on the corresponding
multiplicative quiver variety with a type A surface singularity. We also prove
that the dg versions of these algebras (arising as certain Fukaya categories)
are formal. We conjecture that the same properties hold for all non-Dynkin
quivers, with respect to any extended Dynkin subquiver (note that the cycle is
the type A case). Finally, we prove that multiplicative quiver varieties-for
all quivers-are formally locally isomorphic to ordinary quiver varieties. In
particular, they are all symplectic singularities (which implies they are
normal and have rational Gorenstein singularities). This includes character
varieties of Riemann surfaces with punctures and monodromy conditions. We
deduce this from a more general statement about 2-Calabi--Yau algebras
(following Bocklandt, Galluzzi, and Vaccarino).Comment: 48 pages, Remarks 1.6 and 4.7 adde
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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