Related papers: $L^2$-theory for the $\overline\partial$-operator …

We generalize work of Deligne and Gillet-Soul\'e on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $\Gamma\backslash\mathbb{H}$, for $\Gamma\subset PSL_{2}(\mathbb{R})$ a…

We introduce real-valued $(p,q)$-forms on weighted metric graphs with boundary similar to Lagerberg forms on polyhedral spaces. We compute the Dolbeault cohomology and prove Poincar\'e duality. Using Thuillier's thesis, the skeleton of a…

Let $Y$ be a generic link of a subvariety $X$ of a nonsingular variety $A$. We give a description of the Grauert-Riemenschneider canonical sheaf of $Y$ in terms of the multiplier ideal sheaves associated to $X$ and use it to study the…

Given a bounded constructible complex of sheaves $\mathcal{F}$ on a complex Abelian variety, we prove an equality relating the cohomology jump loci of $\mathcal{F}$ and its singular support. As an application, we identify two subsets of the…

We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential…

We study the weighted spectrum and vanishing cohomology for several classes of isolated hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of…

Let $X$ be a smooth quasi-projective algebraic surface and let $\Delta_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{\Delta_n}$ and their invariants…

In this note we summarize a few of the many recent developments in two-dimensional quantum field theories. We begin with a review of the current state of quantum sheaf cohomology, a heterotic analogue of quantum cohomology. We then turn to…

For a normal subvariety $V$ of ${\bf C}^n$ with a good ${\bf C}^*$-action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the…

Let $\mathcal{H}$ denote the future outgoing null hypersurface emanating from a spacelike 2-sphere $S$ in a vacuum spacetime $(\mathcal{M},\mathbf{g})$. In this paper we study the so-called canonical foliation on $\mathcal{H}$ introduced by…

According to Laumon, an affine Springer fiber is homeomorphic to the universal abelian covering of the compactified Jacobian of a spectral curve. We construct equivariant deformations $f_{n}:\overline{\mathcal{P}}_{n}\to \mathcal{B}_{n}$ of…

When $X=\Gamma\backslash \H^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of $L^2$ harmonic forms vanish. In this paper, we show rigidity…

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this…

Let $M$ be a complete Riemannian manifold satisfying a weighted Poincar\'e inequality, and let $\mathcal{E}$ be a Hermitian vector bundle over $M$ equipped with a metric covariant derivative $\nabla$. We consider the operator…

In the paper, by the singular Riemann-Roch theorem, it is proved that the class of the e-th Frobenius power can be described using the class of the canonical module for a normal local ring of positive characteristic. As a corollary, we…

This work is a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. It is mainly focused on the description of an approach which allows to establish spinorial…

We study the deformation complex of a canonical morphism $i$ from the properad of (degree shifted) Lie bialgebras $\mathbf{Lieb}_{c,d}$ to its polydifferential version $\mathcal{D}(\mathbf{Lieb}_{c,d})$ and show that it is quasi-isomorphic…

We use the anti-equivalence between Cohen-Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S_2 condition…

Let $(X,\gamma)$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\mathrm{dim}(\mathrm{sing}(X))=0$. Let $(F,\tau)\rightarrow X$ be a Hermitian holomorphic vector bundle over $X$ and let us denote with…

The aim of this paper is to establish a canonical decomposition of operator-valued strong $L^2$-functions by the aid of the Beurling-Lax-Halmos Theorem which characterizes the shift-invariant subspaces of vector-valued Hardy space. This…