Related papers: Good minimal models with nowhere vanishing holomor…
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…
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,…
We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold fibres smoothly over the circle, and we give a complete classification of all threefolds with that…
It has been conjectured that varieties of general type do not admit nowhere vanishing holomorphic one-forms. We confirm this conjecture for smooth minimal varieties and for varieties whose Albanese variety is simple.
We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on…
In this note, we prove -- in dimension at most 4 -- a conjectue of Hao which says that a morphism $f : X \to A$ to a simple abelian variety $A$ is smooth if and only if there is a 1-form pulled back from A without any zeros. We also give a…
We prove several conjectures relating the existence of nonvanishing 1- forms to smooth morphisms over abelian varieties, assuming the existence of good minimal models. The proof involves a decomposition result for a family of Calabi-Yau…
We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge D-modules on abelian varieties.
Given a bounded constructible complex of sheaves $\mathcal{F}$ on a complex Abelian variety, we prove an equality relating the cohomology jump loci of $\mathcal{F}$ and its singular support. As an application, we identify two subsets of the…
We prove that a Stein manifold admits a closed holomorphic 1-form without zeros in every class of the first cohomology group. We also prove an approximation result for closed holomorphic 1-forms without zeros defined in a neighborhood of a…
We give a bimeromorphic classification of compact K\"ahler manifolds of Kodaira codimension one that admit a holomorphic one form without zeros.
A conjecture of Kotschick predicts that a compact K\"ahler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in…
We classify all smooth compact connected K\"ahler threefolds that admit the structure of a $C^\infty$-fiber bundle over the circle. This generalizes the work of Hao and Schreieder in the projective case. In contrast to the projective case,…
We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in…
Two conjectures relating the Kodaira dimension of a smooth projective variety and existence of number of nowhere vanishing 1-forms on the variety are proposed and verified in dimension 3.
We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a…
Let $M\stackrel\pi \arrow X$ be a principal elliptic fibration over a Kaehler base $X$. We assume that the Kaehler form on $X$ is lifted to an exact form on $M$ (such fibrations are called positive). Examples of these are regular Vaisman…
A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational…
This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is…
In any dimension at least five we construct examples of closed smooth manifolds with the following properties: 1) they have neither real projective nor flat conformal structures; 2) their fundamental group is a non-elementary Gromov…