Related papers: Perverse obstructions to flat regular compactifica…

Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\Bbb C^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\Bbb…

In the preprint arXiv:2511.07900 we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative…

The introduction is modified in the revised version. Also, many typos and errors were corrected. Let $W\to C$ be degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first constructed…

In this paper, we develop some foundations for a theory of algebraic varieties of congruences on commutative semirings. By studying the structure of congruences, firstly, we show that the spectrum $ \text{Spec}^{c}(A) $ consisting of prime…

We classify $5$-manifolds with fundamental group $\mathbb Z$ and $\pi_{2}$ a finitely generated abelian group in terms of the cup product on the second cohomology of the universal covering. The classification result is applied to study…

Given two nonsingular real algebraic varieties V and W, we consider the problem of deciding whether a smooth map f: V -> W can be approximated by regular maps in the space of smooth maps from V to W. Our main result is a complete solution…

A stratified pseudomanifold is normal if its links are connected. A normalization of a stratified pseudomanifold $X$ is a normal stratified pseudomanifold $Y$ together with a finite-to-one projection $n:Y\to X$ satisfying a local condition…

Keeping N=1 supersymmetry in 4-dimension and in the leading order, we disuss the various orbifold compactifications of M-theory suggested by Horava and Witten on $T^6/Z_3$, $T^6/Z_6$, $T^6/Z_{12}$, and the compactification by keeping…

If P \to X is a topological principal K-bundle and \hat K a central extension of K by Z, then there is a natural obstruction class \delta_1(P) in \check H^2(X,\uline Z) in sheaf cohomology whose vanishing is equivalent to the existence of a…

We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of…

We investigate the formal principle for holomorphic line bundles on neighborhoods of an analytic subset of a complex manifold mainly in the case where it can be realized as an open subset of a compact K\"ahler manifold. Our approach…

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing a certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some…

We consider the problem of supersymmetry breaking in 5 dimensional N=1 supersymmetric models with $S^1/Z_2$ compactification.

We study the affine orbifold laminations that were constructed by Lyubich and Minsky. An important question left open in their construction is whether these laminations are always locally compact. We show that this is not the case. The…

A class of graphs that lies strictly between the classes of graphs of genus (at most) $k-1$ and $k$ is studied. For a fixed orientable surface $S_k$ of genus $k$, let $A_{xy}^k$ be the minor-closed class of graphs with terminals $x$ and $y$…

In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of…

We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli…

Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}\pi{_*}\mathbb{Q}$, where $\pi$ denotes the projection from the universal hyperplane family of…

Let $X$ be a complete $\mathbb{Q}$-factorial toric variety. We explicitly describe the space $H^2(X,T_X)$ and the cup product map $H^1(X,T_X)\times H^1(X,T_X)\to H^2(X,T_X)$ in combinatorial terms. Using this, we give an example of a smooth…