Related papers: Gabriel's theorem and birational geometry

The notion of a semitransitive binary action of a group $G$ on a topological space is introduced. A duality theorem is proved, establishing a bijective correspondence between semitransitive distributive binary $G$-spaces and topological…

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…

We characterise when the Leavitt path algebras over $\mathbb{Z}$ of two arbitrary countable directed graphs are $*$-isomorphic by showing that two Leavitt path algebras over $\mathbb{Z}$ are $*$-isomorphic if and only if the corresponding…

Let T be be a zero-product preserving bounded linear map between C*-algebras A and B. Here neither A nor B is necessarily unital. In this note, we investigate when T gives rise to a Jordan homomorphism. In particular, we show that A and B…

Cohomology of a compatible family of Lie algebroids defined on a family of transverse manifolds is defined. A sheaf of differential forms on a compatible family of Lie algebroids defined over regular open subsets of a simplicial complex is…

We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms $p:X\to B$ are identified with the Hilbert $B$-modules that are equipped with a natural notion of basis. The…

We proved that, in characteristic 0, if two dominant endomorphisms of the projective plane of degree at least 2 are conjugate by some birational transformation, then they are conjugate by an automorphism. We also gave counterexamples in…

Let A^2 be the affine plane over a field K of characteristic 0. Birational morphisms of A^2 are mappings A^2 \to A^2 given by polynomial mappings \phi of the polynomial algebra K[x,y] such that for the quotient fields, one has K(\phi(x),…

Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in…

We prove that a large class of natural transformations (consisting roughly of those constructed via composition from the "functorial" or "base change" transformations) between two functors of the form $\cdots f^* g_* \cdots$ actually has…

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…

The main result says that every surjective isometry between two ideal Banach function spaces satisfying certain conditions can be presented as a composition of a measurable transformation of a variable and multiplication by a function.

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are called {\it $D$-equivalent} if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while {\it…

Let G be a graph with undirected and directed edges. Its representation is given by assigning a vector space to each vertex, a bilinear form on the corresponding vector spaces to each directed edge, and a linear map to each directed edge.…

A conjecture of Amitsur states that two Severi-Brauer varieties are birationally isomorphic if and only if the underlying algebras are the same degree and generate the same cyclic subgroup of the Brauer group. It is known that generating…

We formulate the notion of an isomorphism of GKM graphs. We then show that two GKM graphs have isomorphic graph equivariant cohomology algebras if and only if the graphs are isomorphic.

It is well known that numerical quantities arising from the theory of D-modules are related to invariants of singularities in birational geometry. This paper surveys a deeper relationship between the two areas, where the numerical…

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion…

Let $p\in(1,\infty)\backslash\{2\}$. We show that every homomorphism from a $C^{*}$-algebra $\mathcal{A}$ into $B(l^{p}(J))$ satisfies a compactness property where $J$ is any set. As a consequence, we show that a $C^{*}$-algebra…

We show that the number of birational automorphisms of a variety of general type X is bounded by c \cdot \vol(X,K_X), where c is a constant which only depends on the dimension of X.