Communications in Contemporary Mathematics

The Artin stack ${{ℳ}}_n$ of zero-dimensional sheaves of length $n$ on ${𝔸}^3$ carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack $[crit(f_n)/{\mathrm{\text{GL}}}_n]$, another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points ${\mathrm{\text{Quot}}}_{{𝔸}^3}({\mathcal{𝒪}}^{\oplus r},n)=crit(f_{r,n})$, which is a global critical locus for every $r>0$, and also carries a derived-in-flavor d-critical structure besides the one induced by the potential $f_{r,n}$. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on ${\mathrm{\text{Quot}}}_X(F,n)$, where $F$ is a locally free sheaf of rank $r$ on a projective Calabi–Yau $3$-fold $X$. Finally, we prove that the perfect obstruction theory on ${\mathrm{\text{Hilb}}}^n{𝔸}^3=crit(f_{1,n})$ induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential $f_{1,n}$.

Let $\gamma \in ℝ\setminus \{0\}$ and $X({ℝ}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $\mathrm{\Omega }={ℝ}^n$ or $\mathrm{\Omega }\subset {ℝ}^n$ is an $(𝜀,\infty )$-domain for some $𝜀\in (0,1]$. In this paper, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space ${Ẇ}^{1,X}(\mathrm{\Omega })$ if and only if $f\in L_{\mathrm{\text{loc}}}^1(\mathrm{\Omega })$ and

In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form $u_{xx}$ in usual reaction–diffusion equation, which represent linear diffusion, by terms of form $\varphi {(u)}_{xx}$, representing nonlinear diffusion. We prove the convergence in the fast-reaction limit $k\to \infty$ that is determined by the unique solution of a certain scalar nonlinear diffusion problem.

In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra ${\mathcal{ℛ}}^{\mathrm{\Lambda }}(\beta )$ associated to an arbitrary symmetrizable Cartan matrix $A={(a_{ij})}_{i,j}\in I$, where $\mathrm{\Lambda }\in P^{+}$ and $\beta \in Q_n^{+}$. As applications, we obtain some necessary and sufficient conditions for the KLR idempotent $e(\nu )$ (for any $\nu \in I^{\beta }$) to be nonzero in the cyclotomic quiver Hecke algebra ${\mathcal{ℛ}}^{\mathrm{\Lambda }}(\beta )$. We prove several level reduction results which decompose $dim{\mathcal{ℛ}}^{\mathrm{\Lambda }}(\beta )$ into a sum of some products of $dim{\mathcal{ℛ}}^{{\mathrm{\Lambda }}^i}({\beta }_i)$ with $\mathrm{\Lambda }={\sum }_i{\mathrm{\Lambda }}^i$ and $\beta ={\sum }_i{\beta }_i$, where ${\mathrm{\Lambda }}^i\in P^{+},{\beta }^i\in Q^{+}$ for each $i$. Finally, we construct some explicit monomial bases for the subspaces $e(\tilde{\nu }){\mathcal{ℛ}}^{\mathrm{\Lambda }}(\beta )e(\mu )$ and $e(\mu ){\mathcal{ℛ}}^{\mathrm{\Lambda }}(\beta )e(\tilde{\nu })$ of ${\mathcal{ℛ}}^{\mathrm{\Lambda }}(\beta )$, where $\mu \in I^{\beta }$ is arbitrary and $\tilde{\nu }\in I^{\beta }$ is a certain specific $n$-tuple defined in (5.1).

We describe the prime ideals and, in particular, the maximal ideals in products $R=\prod D_{\lambda }$ of families ${(D_{\lambda })}_{\lambda \in \mathrm{\Lambda }}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $\prod {𝒫}(max(D_{\lambda }))$, where $max(D_{\lambda })$ is the spectrum of maximal ideals of $D_{\lambda }$, and ${𝒫}$ denotes the power set. If every $D_{\lambda }$ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of $R$. If every $D_{\lambda }$ is a Prüfer domain, we completely characterize all prime ideals of $R$.

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and ${𝒪}$-operators are studied.

Let $(Z,\omega )$ be a Kähler manifold and let $U$ be a compact connected Lie group with Lie algebra $𝔲$ acting on $Z$ and preserving $\omega$. We assume that the $U$-action extends holomorphically to an action of the complexified group $U^{ℂ}$ and the $U$-action on $Z$ is Hamiltonian. Then there exists a $U$-equivariant momentum map $\mu :Z\to 𝔲$. If $G\subset U^{ℂ}$ is a closed subgroup such that the Cartan decomposition $U^{ℂ}=U\text{exp}(i𝔲)$ induces a Cartan decomposition $G=K\text{exp}(𝔭),$ where $K=U\cap G$, $𝔭=𝔤\cap i𝔲$ and $𝔤=𝔨\oplus 𝔭$ is the Lie algebra of $G$, there is a corresponding gradient map ${\mu }_{𝔭}:Z\to 𝔭$. If $X$ is a $G$-invariant compact and connected real submanifold of $Z,$ we may consider ${\mu }_{𝔭}$ as a mapping ${\mu }_{𝔭}:X\to 𝔭.$ Given an $\mathrm{\text{Ad}}(K)$-invariant scalar product on $𝔭$, we obtain a Morse like function $f=\frac{1}{2}\parallel {\mu }_{𝔭}{\parallel }^2$ on $X$. We point out that, without the assumption that $X$ is a real analytic manifold, the Lojasiewicz gradient inequality holds for $f$. Therefore, the limit of the negative gradient flow of $f$ exists and it is unique. Moreover, we prove that any $G$-orbit collapses to a single $K$-orbit and two critical points of $f$ which are in the same $G$-orbit belong to the same $K$-orbit. We also investigate convexity properties of the gradient map ${\mu }_{𝔭}$ in the Abelian case. In particular, we study two-orbit variety $X$ and we investigate topological and cohomological properties of $X$.