Related papers: The generalized Lipman-Zariski problem

We prove the Lipman-Zariski conjecture for complex surface singularities with $p_g - g - b \le 2$. Here $p_g$ is the geometric genus, $g$ is the sum of the genera of the exceptional curves and $b$ is the first Betti number of the dual…

Let $X$ be a normal complex space such that the tangent sheaf $T_X$ is locally free and locally admits a basis consisting of pairwise commuting vector fields. Then $X$ is smooth.

We prove the Lipman-Zariski conjecture for complex surface singularities of genus one, and also for those of genus two whose link is not a rational homology sphere. As an application, we characterize complex $2$-tori as the only normal…

Let $X$ be a normal algebraic variety over an algebraically closed field $k$ of characteristic zero. We prove that the K\"{a}hler differential sheaf of $X$ is torsion-free if and only if any regular section of the ideal sheaf of the first…

Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$, logarithmic 1-forms defined on the snc locus of $(X, D)$ extend to a log resolution $\tilde X \to X$ as logarithmic differential forms. We prove that then the…

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha_X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover,…

We give a short proof of the Zariski-Lipman conjecture for toric varieties: any complex toric variety with locally free tangent sheaf is smooth.

We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of a normal complex variety $X$ and an effective Weil divisor $D$ such that the sheaf…

Let $X$ be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of $X$ is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities…

Let $i\colon X\to \Pk^N$ be a projective manifold of dimension $n$ embedded in projective space $\Pk^N$, and let $L$ be the pull-back to $X$ of the line bundle $\Ok_{\Pk^N}(1)$. We construct global explicit Koppelman formulas on $X$ for…

A result of Popa and Schnell shows that any holomorphic 1-form on a smooth complex projective variety of general type admits zeros. More generally, given a variety $X$ which admits $g$ pointwise linearly independent holomorphic 1-forms,…

Let $X$ be a compact K\"ahler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that $X$ admits a Beauville-Bogomolov decomposition: a finite quasi-\'etale cover of $X$ splits as a product…

Let $X$ be a compact normal K\"ahler space whose canonical sheaf is a rank-one free $\mathcal O_X$ module and whose singularities are isolated, rational and quasi-homogeneous. We prove then that under a topological hypothesis the…

We study the main open parts of the Kawaguchi--Silverman Conjecture, asserting that for a birational self-map $f$ of a smooth projective variety $X$ defined over $\overline{\mathbb Q}$, the arithmetic degree $\alpha_f(x)$ exists and…

Based on the celebrated result on zeros of holomorphic 1-forms on complex varieties of general type by Popa and Schnell, we study holomorphic 1-forms on $n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complex minimal…

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial…

In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion free. In addition, when the codimension of the singular locus is at least two, it is shown that being…

In this note, we show that if $f\colon M\rightarrow X$ is a germ of a projective Lagrangian fibration from a holomorphic symplectic manifold $M$ onto a normal analytic variety $X$ with isolated quotient singularities, then $X$ is smooth. In…

We prove that if $X$ is the total space of an elliptic principal bundle $\pi:X\ra B$ which is non-K\"ahler, then the restriction of any torsion-free sheaf on $X$ to the general fiber of $\pi$ is semi-stable.

Let $X$ be a normal compact K\"ahler space with klt singularities and torsion canonical bundle. We show that $X$ admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then…