arXiv subjects

Martin Schwald

Publications and source records attributed to Martin Schwald.

7 recordsLinked to original sources

arXiv2023

Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperk\"ahler metrics induced by deformed twistor cycles

We answer a question posed independently by Fels-Huckleberry-Wolf and Looijenga concerning the geometric meaning of small deformations of twistor cycles in the K3 period domain. These are shown to induce complex-hyperk\"ahler metrics on members of the families via Penrose's Non-linear Graviton construction. On the way to proving this result, we construct a Hausdorff fine moduli space for families of marked K3 surfaces over smooth rational curves in the K3 period domain. Over an open subset containing all twistor cycles we construct a family of such families, which is a universal small deformation for every twistor family. Whenever possible, we extend the results to higher-dimensional irreducible holomorphic-symplectic manifolds.

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arXiv2021

The Kodaira problem for Kähler spaces with vanishing first Chern class

Let $X$ be a normal compact Kähler 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 prove that this unobstructedness assumption holds in at least three cases: if $X$ has toroidal singularities, if $X$ has finite quotient singularities, and if the second cohomology group of its tangent sheaf vanishes.

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arXiv2020

On the Kodaira problem for uniruled Kähler spaces

We discuss the Kodaira problem for uniruled Kähler spaces. Building on a construction due to Voisin, we give an example of a uniruled Kähler space $X$ such that every run of the $K_X$-MMP immediately terminates with a Mori fibre space, yet $X$ does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.

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arXiv2020

On the definition of irreducible holomorphic symplectic manifolds and their singular analogs

In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss finite quotients X of complex tori where the space of reflexive 2-forms is generated by a holomorphic symplectic form, and their Lagrangian fibrations. Neither X nor the base can be smooth unless X is a 2-torus.

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arXiv2018

Unobstructedness of hyperkähler twistor spaces

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

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arXiv2017

Fujiki relations and fibrations of irreducible symplectic varieties

This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has the same rank as in the smooth case. This enables us to study fibrations of these varieties; imposing the newer definition from [GKP16, Definition 8.16.2] we show that they behave much like irreducible symplectic manifolds.

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arXiv2016

Low degree Hodge theory for klt varieties

If X is a complex projective variety with klt singularities, then the mixed Hodge structures on the first two singular cohomology groups are pure. We describe the pieces of the Hodge decomposition in terms of reflexive differential forms. Applications include a Lefschetz (1,1) Theorem and a weak analogue of the Hodge-Riemann bilinear relations for klt varieties.

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