Related papers: MacPherson's Conjecture via H\"ormander Estimate

We give a proof of the openness conjecture of Demailly and Koll\'ar for positively curved singular metrics on ample line bundles over projective varieties. As a corollary it follows that the openness conjecture for plurisubharmonic…

This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and…

We give an elementary algebraic proof of some asymptotic estimates (called by Demailly asymptotic Morse inequalities) for the dimensions of cohomology groups of the difference of two ample line bundles on a smooth complex projective variety…

The celebrated Ohsawa--Takegoshi extension theorem for $L^2$ holomorphic functions on bounded pseudoconvex domains in $\mathbb C^n$ is a fundamental result in several complex variables and complex geometry. Ohsawa conjectured in 1995 that…

We obtain a priori local pointwise second derivative estimates for solutions $u$ to a class of augmented Hessian equations on Riemannian manifolds, in terms of the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case that no…

In this article, we pursue the study begun in \cite{Lup02} on the cohomology of rationally elliptic coformal spaces. Consequently, we complete, for such spaces, the proof of Lupton's conjecture and deduce Hilali's.

Hermite reciprocity refers to a series of natural isomorphisms involving compositions of symmetric, exterior, and divided powers of the standard $SL_2$-representation. We survey several equivalent constructions of these isomorphisms, as…

We state a precise conjectural isomorphism between localizations of the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian, in particular relating their Schubert bases and…

We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence…

We explain how Teleman quantization can be applied to moduli spaces of quiver representations to compute the higher cohomology of the endomorphism bundle of the universal bundle. We use this to prove Schofield's partial tilting conjecture,…

We introduce the \verb|Macaulay2| package \verb|RepHomology| for the computations of representation homology of certain spaces. The main methods implement computing the representation homology of surfaces (with group coefficients, and…

We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing…

The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each…

We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight…

A classical theorem of Frankel for compact K\"ahler manifolds states that a K\"ahler S^1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when Hodge theory holds on non-compact manifolds, then…

Not long ago, Cirici and Wilson defined a Dolbeault cohomology on almost complex manifolds to answer Hirzebruch's problem. In this paper, we define a refined Dolbeault cohomology on almost complex manifolds. We show that the condition…

We study the behavior of cohomological support loci of the canonical bundle under derived equivalence of smooth projective varieties. This is achieved by investigating the derived invariance of a generalized version of Hochschild homology.…

The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's…

We establish a new generalization of an $L^2$ extension theorem of Ohsawa-Takegoshi type. The improvement in the theorem is that it allows the usual curvature assumptions to be significantly weakened in certain favorable settings. The…

Manin's conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety in terms of its global geometric invariants. The strongest form of the conjecture implies certain…