Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

We study the motivic and $$\ell $$ ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

$ \bar{\partial} $-equation look at analytic Hilbert's zero-locus theorem

<abstract><p>Stemming from the Pythagorean Identity $ \sin^2z+\cos^2z = 1 $ and Hörmander's $ L^2 $-solution of the Cauchy-Riemann's equation $ \bar{\partial}u = f $ on $ \mathbb C $, this article demonstrates a corona-type principle which exists as a somewhat unexpected extension of the analytic Hilbert's Nullstellensatz on $ \mathbb C $ to the quadratic Fock-Sobolev spaces on $ \mathbb C $.</p></abstract>

$ \bar{\partial} $-equation look at analytic Hilbert's zero-locus theorem

<abstract><p>Stemming from the Pythagorean Identity $ \sin^2z+\cos^2z = 1 $ and Hörmander's $ L^2 $-solution of the Cauchy-Riemann's equation $ \bar{\partial}u = f $ on $ \mathbb C $, this article demonstrates a corona-type principle which exists as a somewhat unexpected extension of the analytic Hilbert's Nullstellensatz on $ \mathbb C $ to the quadratic Fock-Sobolev spaces on $ \mathbb C $.</p></abstract>

Semiorthogonal Decompositions on Total Spaces of Tautological Bundles

Let $U$ be the tautological subbundle on the Grassmannian $\operatorname{Gr}(k, n)$. There is a natural morphism $\textrm{Tot}(U) \to{\mathbb{A}}^n$. Using it, we give a semiorthogonal decomposition for the bounded derived category $D^b_{\!\textrm{coh}}(\textrm{Tot}(U))$ into several exceptional objects and several copies of $D^b_{\!\textrm{coh}}({\mathbb{A}}^n)$. We also prove a global version of this result: given a vector bundle $E$ with a regular section $s$, consider a subvariety of the relative Grassmannian $\operatorname{Gr}(k, E)$ of those subspaces that contain the value of $s$. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of $s$. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case $k = 1$.

Galois level and congruence ideal for -adic families of finite slope Siegel modular forms

We consider families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for $\operatorname{GL}_{2}$, via a $p$-adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the $p$-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for $\operatorname{GL}_{2}$ to an eigenvariety for $\operatorname{GSp}_{4}$, while the remainder appear as isolated points on the eigenvariety.

Shimura Varieties: A Hodge-Theoretic Perspective

This chapter discusses certain cleverly constructed unions of modular varieties, called Shimura varieties, in the Hodge-theoretic perspective. The Shimura varieties can show the minimal (i.e., reflex) field of definition of a Hodge/zero locus setting, and also reveal quite a bit about the interplay between “upstairs” and “downstairs” (in Ď and Γ‎\D, respectively) fields of definition of subvarieties. Hence, the chapter defines the Hermitian symmetric domains in D as well as the locally symmetric varieties Γ‎\D. It then discusses the theory of complex multiplication, before introducing Shimura varieties ∐ᵢ(Γ‎ᵢ\D) as well as three key Adélic lemmas, before finally laying out the fields of definition.

Quantum 𝒟-modules for toric nef complete intersections

Let [Formula: see text] be a smooth projective toric variety with [Formula: see text] ample line bundles. Let [Formula: see text] be the zero locus of [Formula: see text] generic sections. It is well known that the ambient quantum [Formula: see text]-module of [Formula: see text] is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of [Formula: see text] and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum [Formula: see text]-module.