Related papers: Multiplicity and the pull-back problem

Let $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$, where $\kappa=(k_1,\dots,k_n)$, be a stratum of (projectivized) $d$-differentials in genus $0$. We prove a recursive formula which relates the volume of…

In this paper, we prove that the foliated Rosenberg index of a possibly noncompactly enlargeable, spin foliation is nonzero. It generalizes our previous result. The difficulty brought by the noncompactness is reflected in the infinite…

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed…

A classical theorem of Wonenburger, Djokovic, Hoffmann and Paige states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give…

This paper is devoted to the study of the LNE property in complex analytic hypersurface parametrized germs, that is, the sets that are images of finite analytic map germs from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$. We prove that if…

Suppose $\pi:W\to S$ is a smooth, proper morphism over a variety $S$ contained as a Zariski open subset in a smooth, complex variety $\bar S$. The goal of this note is to consider the question of when $\pi$ admits a regular, flat…

In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_{\lambda}) in a smooth bounded domain, driven by a nonlocal integrodifferential operator…

Let $\{f_{\lambda; j}\}_{\lambda\in V; 1\le j\le k}$ be families of holomorphic functions in the open unit disk $\Di\subset\Co$ depending holomorphically on a parameter $\lambda\in V\subset \Co^n$. We establish a Rolle type theorem for the…

In this paper we are interested in the fine-grained complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$ (the $H$-Coloring problem). The starting point is that these problems can be viewed as…

We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas's work on integral Grothendieck-Riemann-Roch. If $S$ is smooth quasi-projective of dimension $d$ over a field and $\pi \colon X\to…

We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by…

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based…

We prove a Lojasiewicz-Simon inequality $$ \left| E(u) - 4\pi n \right| \leq C \| \mathcal{T}(u) \|^\alpha $$ for maps $u \in W^{2,2}\left( S^2, S^2 \right).$ The inequality holds with $\alpha = 1$ in general and with $\alpha > 1$ unless…

We find conditions which ensure that the topological complexity of a closed manifold $M$ with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on…

We prove a query complexity variant of the weak polynomial Freiman-Ruzsa conjecture in the following form. For any $\epsilon > 0$, a set $A \subset \mathbb{Z}^d$ with doubling $K$ has a subset of size at least $K^{-\frac{4}{\epsilon}}|A|$…

We prove an algebraic formula, conjectured by M. Kontsevich, for computing the monodromy of the vanishing cycles of a regular function on a smooth complex algebraic variety.

Let $f : X \to S$ be a smooth projective family defined over $\mathcal{O}_{K}[\mathcal{S}^{-1}]$, where $K \subset \mathbb{C}$ is a number field and $\mathcal{S}$ is a finite set of primes. For each prime $\mathfrak{p} \in…

Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field…

Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ be finite…

We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called…