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Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we show how Satos specialization and microlocalization functors have a natural enhancement, and discuss some of their properties.
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Let X be a smooth, connected, closed subvariety of a complex vector space V. The asymptotic cone as(X) is naturally equipped with a nearby cycles sheaf P coming from the specialization of X to as(X). We show that if X is transverse to infinity in a suitable sense, then the Fourier transform of P is an intersection homology sheaf.
We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel to prove the specialization of birational types. We also provide several explicit examples of obstructions to stable rationality arising from this technique.
Let $R$ be a commutative noetherian ring, and $mathcal{Z}$ a stable under specialization subset of $Spec(R)$. We introduce a notion of $mathcal{Z}$-cofiniteness and study its main properties. In the case $dim(mathcal{Z})leq 1$, or $dim(R)leq 2$, or $R$ is semilocal with $cd(mathcal{Z},R) leq 1$, we show that the category of $mathcal{Z}$-cofinite $R$-modules is abelian. Also, in each of these cases, we prove that the local cohomology module $H^{i}_{mathcal{Z}}(X)$ is $mathcal{Z}$-cofinite for every homologically left-bounded $R$-complex $X$ whose homology modules are finitely generated and every $i in mathbb{Z}$.
Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we give some precisions on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $mathcal M$ be a holonomic algebraic $mathcal D$-module on the affine line, and denote by ${}^{mathsf{L}}mathcal M$ its Fourier-Laplace transform. For a point $a$ on the affine line, denote by $ell_a$ the corresponding linear function on the dual affine line. Then, the vanishing cycles of $mathcal M$ at $a$ are isomorphic to the graded component of degree $ell_a$ of the Stokes filtration of ${}^{mathsf{L}}mathcal M$ at infinity.
We discuss a model of an economic community consisting of $N$ interacting agents. The state of each agent at any time is characterized, in general, by a mixed strategy profile drawn from a space of $s$ pure strategies. The community evolves as agents update their strategy profiles in response to payoffs received from other agents. The evolution equation is a generalization of the replicator equation. We argue that when $N$ is sufficiently large and the payoff matrix elements satisfy suitable inequalities, the community evolves to retain the full diversity of available strategies even as individual agents specialize to pure strategies.
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