On a singular variety associated to a polynomial mapping

In the paper "Geometry of polynomial mapping at infinity via intersection homology" the second and third authors associated to a given polynomial mapping F : \C^2 \to \C^2 with nonvanishing jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of the mapping. We generalize this result.Comment: 1 figur

A New Large N Expansion for General Matrix-Tensor Models

We define a new large $N$ limit for general $\text{O}(N)^{R}$ or $\text{U}(N)^{R}$ invariant tensor models, based on an enhanced large $N$ scaling of the coupling constants. The resulting large $N$ expansion is organized in terms of a half-integer associated with Feynman graphs that we call the index. This index has a natural interpretation in terms of the many matrix models embedded in the tensor model. Our new scaling can be shown to be optimal for a wide class of non-melonic interactions, which includes all the maximally single-trace terms. Our construction allows to define a new large $D$ expansion of the sum over diagrams of fixed genus in matrix models with an additional $\text{O}(D)^{r}$ global symmetry. When the interaction is the complete vertex of order $R+1$, we identify in detail the leading order graphs for $R$ a prime number. This slightly surprising condition is equivalent to the complete interaction being maximally single-trace.Comment: 57 pages, 20 figures (additional discussion in Sec. 4.1.1. and additional figure (Fig. 5)

Poincaré duality for LpL^{p} cohomology on subanalytic singular spaces

We investigate the problem of Poincaré duality for $L^{p}$ differential forms on bounded subanalytic submanifolds of Rn (not necessarily compact). We show that, when p is sufficiently close to 1 then the $L^{p}$ cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that Lp cohomology is dual to intersection homology. As a consequence, we can deduce that the $L^{p}$ cohomology is Poincaré dual to Lq cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large

L∞L^\infty cohomology is intersection cohomology

Let $X$ be any subanalytic compact pseudomanifold. We show a De Rham theorem for $L^\infty$ forms. We prove that the cohomology of $L^\infty$ forms is isomorphic to intersection cohomology in the maximal perversity.Comment: Appendix on subanalytic geometry adde

A Lefschetz duality for intersection homology

We prove a Lefschetz duality theorem for intersection homology. Usually, this result applies to pseudomanifolds with boundary which are assumed to have a “collared neighborhood of their boundary”. Our duality does not need this assumption and is a generalization of the classical one