For a subanalytic Legendrian \(\Lambda \subseteq S^{*}M\), we prove that when \(\Lambda\) is either swappable or a full Legendrian stop, the microlocalization at infinity \(m_\Lambda: ...\operatorname{Sh}_\Lambda(M) \rightarrow \operatorname{\mu sh}_\Lambda(\Lambda)\) is a spherical functor, and the spherical cotwist is the Serre functor on the subcategory \(\operatorname{Sh}_\Lambda^b(M)_0\) of compactly supported sheaves with perfect stalks. In this case, when \(M\) is compact the Verdier duality on \(\operatorname{Sh}_\Lambda^b(M)\) extends naturally to all compact objects \(\operatorname{Sh}_\Lambda^c(M)\). This is a sheaf theory counterpart (with weaker assumptions) of the results on the cap functor and cup functors between Fukaya categories. When proving spherical adjunction, we deduce the Sato-Sabloff fiber sequence and construct the Guillermou doubling functor for any Reeb flow. As a setup for the Verdier duality statement, we study the dualizability of \(\operatorname{Sh}_\Lambda(M)\) itself and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels.
Let
X
be a projective K3 surfaces. In two examples where there exists a fine moduli space
M
of stable vector bundles on
X
, isomorphic to a Hilbert scheme of points, we prove that the universal ...family
E
on
X
×
M
can be understood as a complete flat family of stable vector bundles on
M
parametrized by
X
, which identifies
X
with a smooth connected component of some moduli space of stable sheaves on
M
.
Constructible sheaves on schemes Hemo, Tamir; Richarz, Timo; Scholbach, Jakob
Advances in mathematics (New York. 1965),
09/2023, Volume:
429
Journal Article
Peer reviewed
Open access
We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable ...objects in the derived ∞-category of proétale sheaves, while constructible sheaves are those that are lisse on a stratification.
We show that constructible sheaves satisfy proétale descent. We also establish a t-structure on constructible sheaves in a wide range of cases. We finally provide a toolset to manipulate categories of constructible sheaves with respect to the choices of coefficient rings, and use this to prove that our notions reproduce and extend the various approaches to, say, constructible ℓ-adic sheaves in the literature.
We review how phase-field models contributed to the understanding of various aspects of crystal nucleation, including homogeneous and heterogeneous processes, and their role in microstructure ...evolution. We recall results obtained both by the conventional phase-field approaches that rely on spatially averaged (coarse grained) order parameters in capturing freezing, and by the recently developed phase-field crystal models that work on the molecular scale, while employing time averaged particle densities, and are regarded as simple dynamical density functional theories of classical particles. Besides simpler cases of homogeneous and heterogeneous nucleation, phenomena addressed by these techniques include precursor assisted nucleation, nucleation in eutectic and phase separating systems, phase selection via competing nucleation processes, growth front nucleation (a process, in which grains of new orientations form at the solidification front) yielding crystal sheaves and spherulites, and transition between the growth controlled cellular and the nucleation dominated equiaxial solidification morphologies.
Part One of this book covers the abstract foundations of Grothendieck duality theory for schemes in part with noetherian hypotheses and with some refinements for maps of finite tor-dimension. Part ...Two extends the theory to the context of diagrams of schemes.
We introduce the notion of mixed-\(\omega\)-sheaves and use it for the study of a relative version of Fujita's freeness conjecture. It is related to the Iitaka conjecture. We note that the notion of ...mixed-\(\omega\)-sheaves is a generalization of that of Nakayama's \(\omega\)-sheaves in some sense. One of the main motivations of this paper is to make Nakayama's theory of \(\omega\)-sheaves more accessible and make it applicable to the study of log canonical pairs.
We introduce the notion of Chern-Simons classes for curved DG-pairs and we prove that a particular case of this general construction provides canonical \(L_\infty\) liftings of Buchweitz-Flenner ...semiregularity maps for coherent sheaves on complex manifolds.
Abstract
We study log
$\mathscr {D}$
-modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized
b
-functions. As applications, we ...deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.
We consider the algebra of Hecke correspondences (elementary transformations at a single point) acting on the algebraic
K
-theory groups of the moduli spaces of stable sheaves on a smooth projective ...surface
S
. We derive quadratic relations between the Hecke correspondences, and compare the algebra they generate with the Ding–Iohara–Miki algebra (at a suitable specialization of parameters), as well as with a generalized shuffle algebra.
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