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Each compact Riemannian manifold with no conjugate points admits a family of functions whose integrals vanish exactly when central Busemann functions split linearly. These functions vanish when all central Busemann functions are sub- or…

We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic.

Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k'$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre…

We consider the vector space $E_{\rho,p}$ of entire functions of finite order, whose types are not more than $p>0$, endowed with Frechet topology, which is generated by a sequence of weighted norms. We call a function $f\in E_{\rho,p}$ {\it…

The behavior near the boundary in the Deligne-Mumford compactification of many functions on the moduli space of pointed Riemann surfaces can be conveniently expressed using the notion of "point-like limit" that we adopt from the string…

We prove an algebraic decomposition theorem for the unit group $\mathrm{GL}(R)$ of an arbitrary non-discrete irreducible, continuous ring $R$ (in von Neumann's sense), which entails that every element of $\mathrm{GL}(R)$ is both a product…

Let $X, Y$ be two complex manifolds, let $D\subset X,$ $ G\subset Y$ be two nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup…

We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither…

We prove that every orientable infinite type surface without boundary and finite genus has a Riemann surface structure such that its modular group of quasiconformal homeomorphisms is countable.

Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN in P implies P=NP and, thanks to recent work of Koiran, it is now known that…

Let S be a generic C-infinity smooth CR manifold in C^n, n > 1, and let M be a generic C-infinity CR submanifold of S X C^m. We prescribe conditions on M so that it is the disjoint union of graphs of CR maps f:S-->C^m. We also consider the…

We prove the conjectures of Yau-Zaslow and Gottsche concerning the number curves on K3 surfaces. Specifically, let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of…

Let X be a finite CW complex or compact Lipschitz neighborhood retract with universal cover Z; let M be a compact orientable manifold of dimension at least 2 and nonempty boundary. We establish the existence of an isoperimetric profile for…

For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banach algebras $\mathcal H^\infty(B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra $\mathcal A_u(B_X)$ of uniformly continuous functions on…

Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can…

We classify all finite subgroups of the plane Cremona group which have a fixed point. In other words, we determine all rational surfaces X with an action of a finite group G such that X is equivariantly birational to a surface which has a…

In this note we show that on any compact subdomain of a K\"ahler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized…

Let $k$ be a field of characteristic zero. By using Hironaka's desingularisation theorem, we prove an extension criterion for a functor defined on nonsingular k-schemes and taking values on a category of complexes. Roughly speaking, the…

We prove that an infinite Riemann surface $X$ is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are cross-cuts is of zero measure. Then we establish…

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate…