Related papers: The algebraic dimension of compact complex threefo…

In this paper, we study numerically flat holomorphic vector bundles over a compact non-K\"ahler manifold $(X, \omega)$ with the Hermitian metric $\omega$ satisfying the Gauduchon and Astheno-K\"ahler conditions. We prove that numerically…

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…

Let $X$ be any smooth prime Fano threefold of degree $2g-2$ in $\mathbb{P}^{g+1}$, with $g \in \{3,\ldots,10,12\}$. We prove that for any integer $d$ satisfying $\left\lfloor \frac{g+3}{2} \right\rfloor \leq d \leq g+3$ the Hilbert scheme…

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects…

We give an explicit presentation of a family of finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero that constitute all liftings of Nichols algebras of diagonal Cartan type $B_{3}$ over a…

This paper is about geometric and topological properties of a proper CAT(0) space $X$ which is cocompact - i.e. which has a compact generating domain with respect to the full isometry group. It is shown that geodesic segments in $X$ can…

The question of whether a closed, orientable manifold can admit a nontrivial vector field that is parallel with respect to some Riemannian metric is a classical problem in Differential Geometry, first posed by S. S. Chern [11]. In this…

We prove Grothendieck's Conjecture on Resolution of Singulari-ties for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = SpecA, A a reduced complete Noetherian local ring of…

In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting.…

We introduce notions of {\it upper chernrank} and {\it even cup length} of a finite connected CW-complex and prove that {\it upper chernrank} is a homotopy invariant. It turns out that determination of {\it upper chernrank} of a space $X$…

This thesis is devoted to the study of abelian automorphism groups of surfaces and $3$-folds of general type over complex number field $\Bbb C$. We obtain a linear bound in $K^3$ for abelian automorphism groups of $3$-folds of general type…

Motivated by the path integral analysis of boundary conditions in a 3-dimensional topological sigma-model, we suggest a definition of the 2-category associated with a holomorphic symplectic manifold X and study its properties. The simplest…

Let $n\geq 2$ and $r\in \{1, \cdots, n-1\}$ be integers, $M$ be a compact smooth K\''ahler manifold of complex dimension $n$, $E$ be a holomorphic vector bundle with complex rank $r$ and equipped with an hermitian metric $h_E$, and $L$ be…

An exact computation of the persistent Betti numbers of a submanifold $X$ of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of $X$ is available. We show that, under suitable…

We prove that the zeroth L^2-Betti number of a compact quantum group vanishes unless the underlying C*-algebra is finite dimensional and that the zeroth L^2-homology itself is non-trivial exactly when the quantum group is coamenable.

Let $X$ be a non-degenerate projective irreducible variety of dimension $n \ge 1$, degree $d$, and codimension $e \ge 2$ over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Let $\beta_{p,q} (X)$ be the $(p,q)$-th graded…

The full solenoid over a topological space $X$ is the inverse limit of all finite covers. When $X$ is a compact Hausdorff space admitting a locally path connected universal cover, we relate the pointed homotopy equivalences of the full…

We introduce a natural class of models of random chain complexes of real vector spaces that some classical ensembles of random matrices, the length $1$ case. We are interested here in the homological properties of these random complexes.…

It is known that the optimal Noether inequality $\mathrm{vol}(X) \ge \frac{4}{3}p_g(X) - \frac{10}{3}$ holds for every $3$-fold $X$ of general type with $p_g(X) \ge 11$. In this paper, we give a complete classification of $3$-folds $X$ of…

We discuss notions of almost complex, complex and K\"{a}hler structures in the realm of non-commutative geometry and investigate them for a class of finite dimensional spectral triples on the three-point space. We classify all the almost…