The graded coherent sheaf $α_X^\bullet$ constructed in [B.18] for any reduced pure dimensional complex space $X$ is stable by exterior product but not by the de Rham differential. We construct here a new graded coherent sheaf $α_X^\bullet$ containing $α_X^\bullet$ and stable both by exterior product and by the de Rham differential. We show that it has again the ``pull-back property'' for holomorphic maps $f : X \to Y$ between irreducible complex spaces such that $f(X)$ is not contained in the singular set of $Y$. Moreover, this graded coherent sheaf $α_X^\bullet$ comes with a natural coherent exhaustive filtration and this filtration is also compatible with the pull-back by such holomorphic maps. These sheaves define new invariants on singular complex spaces. We show on some simple examples that these invariants are new.
We introduce in a reduced complex space, a "new coherent sub-sheaf" of the sheaf $ω\_{X}^{\bullet}$ which has the "universal pull-back property" for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms $Ω\_{X}^{\bullet}/torsion$. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf $Ω\_{X}^{\bullet}/torsion$. This sheaf $α\_{X}^{\bullet}$ is also closely related to the normalized Nash transform. We also show that these $q-$meromorphic d…
On an $n$-dimensional locally reduced complex analytic space $X$ on which the shifted constant sheaf $\Q_X^\bullet[n]$ is perverse, it is well-known that, locally, $\Q_X^\bullet[n]$ underlies a mixed Hodge module of weight $\leq n$ on $X$, with weight $n$ graded piece isomorphic to the intersection cohomology complex $\Idot_X$ with constant $\Q$ coefficients. In this paper, we identify the weight $n-1$ graded piece $\Gr_{n-1}^W \Q_X^\bullet[n]$ in the case where $X$ is a "parameterized space", using the comparison complex, a perverse sheaf naturally defined on any space for which the shifted c…
In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof of Serre duality on any reduced pure $n$-dimensional paracompact complex space $X$. At the core of the paper is the introduction of concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a comple…
The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex …
Raynaud and Gruson developed the theory of blowing-up an algebraic variety $X$ along a coherent sheaf $M$ in the sense that there exists a blow-up $X'$ of $X$ such that the "strict transform" of $M$ is flat over $X'$ and the blow-up satisfies an universal (minimality) property. However, not much is known about the singularities of the blow-up. In this article, we prove that if $X$ is a normal surface singularity and $M$ is a reflexive $\mathcal{O}_{X}$-module, then such a blow-up arises naturally from the theory of McKay correspondence. We show that the normalization of the blow-up of Raynaud …
We construct a generalization of the Dolbeault-Grothendieck resolution on a singular complex space. The same construction yields, for each morphism of analytic spaces, a pullback mapping between the respective Dolbeault-Grothendieck resolutions. As in the smooth case, the terms of the resolution are soft sheaves with stalks which are flat with respect to the sheaf of holomorphic sections. If, moreover, the complex space $(X,\mathcal{O}_{X})$ is countable at infinity then the global section spaces of the terms of the resolution are endowed with natural Fréchet-Schwarz topologies which induce th…
Let $X$ be a complex smooth projective variety, and $\mathcal{G}$ a locally free sheaf on $X$. We show that there is a 1-to-1 correspondence between pairs $(Λ,Ξ)$, where $Λ$ is a sheaf of almost polynomial filtered algebras over $X$ satisfying Simpson's axioms and $Ξ: \GrΛ\rightarrow \Sym^\bullet_{\corO_X} \mathcal{G}$ is an isomorphism, and pairs $(\mathcal{L},Σ)$, where $\mathcal{L}$ is a holomorphic Lie algebroid structure on $\mathcal{G}$ and $Σ$ is a class in $F^1H^2(\mathcal{L},\C)$, the first Hodge filtration piece of the second cohomology of $\bella$. As an application, we construct mo…
For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at infinity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor…
We propose and study a generalized version of the Lipman-Zariski conjecture: let $(x \in X)$ be an $n$-dimensional singularity such that for some integer $1 \le p \le n - 1$, the sheaf $Ω_X^{[p]}$ of reflexive differential $p$-forms is free. Does this imply that $(x \in X)$ is smooth? We give an example showing that the answer is no even for $p = 2$ and $X$ a terminal threefold. However, we prove that if $p = n - 1$, then there are only finitely many log canonical counterexamples in each dimension, and all of these are isolated and terminal. As an application, we show that if $X$ is a projecti…
We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and equip it with a natural paracyclic structure (``rotational symmetry''). This fulfills a part of the general program to provide a rigorous account of perverse schobers which are (thus far conjectural) categorifications of perverse sheaves. Namely, in terms of our previous identification of perverse sheaves on Riemann surfaces with Milnor sheaves, the relative …
Let $f: X \to S$ be a unipotent degeneration of projective complex manifolds over a disc such that the reduction of the central fibre $Y=f^{-1}(0)$ is simple normal crossings, and let $X_\infty$ be the canonical nearby fibre. Building on the work of Kontsevich, Tschinkel, Mikhalkin and Zharkov, I introduce a sheaf of graded algebras $Λ^\bullet$ on the dual intersection complex of $Y$, denoted $Δ_X$. I show that there exists a map $H^q(Δ_X, Λ^p) \to \mathrm{gr}^W_{2p} H^{p+q}(X_\infty, \mathbb{Q})$, where $W$ is the monodromy weight filtration, which is injective whenever there exists a class $…
We introduce a new metric homology theory, Moderately Discontinuous Homology, which captures Lipschitz properties of metric subanalytic germs. The main novelty is to allow "moderately discontinuous" chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group $MDH^b_\bullet$ for any $b\in [1,\infty]$ and homomorphisms $MDH^b_\bullet\to MDH^{b'}_\bullet$ for any $b\geq b'$. Here $b$ is a "discontinuity rate". The homology groups for the inner or outer metric are proved to be finitely generated and…
This paper studies formality of the differential graded algebra $RHom(E,E)$, where $E$ is a semistable sheaf on a K3 surface. The main tool is Kaledin's theorem on formality in families. For a large class of sheaves $E$, this DG algebra is formal, therefore we have an explicit description of the singularity type of the moduli space of semistable sheaves at the point represented by $E$. This paper also explains why Kaledin's theorem fails to apply in the remaining case.
