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Stefan Schreieder

Publications and source records attributed to Stefan Schreieder.

At least 19 recordsLinked to original sources
arXiv2026

The topology of Kähler manifolds and 1-forms without zeros

We study how the topology and geometry of a compact Kähler manifold relate to the zeros of its one-forms. We determine all implications among several closely related conditions. In particular, we construct a smooth projective variety for which the Aomoto complex is exact for every nonzero holomorphic one-form and every semisimple local system, although every closed real one-form, and hence every holomorphic one-form, has a zero.

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arXiv2026

Invisible singularities in complex algebraic geometry

We construct morphisms between smooth complex projective varieties that have singular fibers, but look topologically smooth. We use this to give negative answers to the following four conjectures and questions: the smoothness conjecture of Fernández~de~Bobadilla and Kollár, a question of Kollár and Pardon on universal covers, Kotschick's conjecture on 1-forms, and a conjecture of Schreieder on Aomoto complexes. Version 2 is a revised version of sections 1--6 of the original, no change in the results. A substantially expanded version of sections 7--8 of the original will be posted separately.

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arXiv2026

Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces

The Cassels--Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray (1976) and Voisin (2026) to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels--Swinnerton-Dyer conjecture for cubic surfaces.

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arXiv2026

Zeros of one-forms and the topology of algebraic maps

We construct a smooth complex projective variety whose Albanese morphism is a homotopy fiber bundle but not a submersion. The same variety fibers smoothly over the circle, although every holomorphic one-form on it has a zero. A second construction yields smooth complex projective varieties $X$ such that the Aomoto complex of every nonzero holomorphic one-form on every connected finite étale cover of $X$ is exact, while $X$ admits no real closed one-form without zeros. The two constructions build, respectively, on a homology fiber bundle of Corrêa--Kollár that is not a homotopy fiber bundle and on a rational cohomology torus constructed by Debarre--Jiang--Lahoz. Consequently, we disprove Kotschick's conjecture, the remaining implication in the Bobadilla--Kollár conjecture, and a conjecture of the first-named author.

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arXiv2026

On the degree of subvarieties on abelian varieties

Let $(X,Θ)$ be a very general principally polarized abelian variety of dimension $g$, and consider the minimal cohomology class $θ_k=[Θ]^k/k!$ for $k<g$. We show that the minimal positive multiple of $θ_k$ which is algebraic is divisible by all primes $p\leq (k+1)/2$. In particular, these minimal multiples grow exponentially with $k$. Our main result follows from [EGFS25] together with a new combinatorial result about $\mathbb F_p$-solutions of certain graphic matroids in their own Albanese graphs.

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arXiv2026

Divisibility phenomena in motivic Bloch--Ogus theory

Let X be a smooth projective variety over a field k. For k separably closed, we prove that the subgroup of unramified classes in the Milnor K-group $K^M_i(k(X))$ of the function field of X is contained in the subgroup of n-divisible elements of $K^M_i(k(X))$ for any integer n invertible in k. This generalizes to a statement for unramified motivic cohomology of arbitrary bidegree. We further show that whenever k is finite or separably closed and l is a prime invertible in k, then all but the last step in the Bloch--Ogus filtration of the motivic cohomology of X are l-divisible up to torsion. Generalizations of this last result to arbitrary quasi-projective k-schemes are also proven.

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arXiv2026

Rationality of hypersurfaces

We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.

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arXiv2026

Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction

We survey the Mumford construction of degenerating abelian varieties, with a focus on the analytic version of the construction, and its relation to toric geometry. Moreover, we study the geometry and Hodge theory of multivariable degenerations of abelian varieties associated to regular matroids, and extend some fundamental results of Clemens on 1-parameter semistable degenerations to the multivariable setting.

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arXiv2026

Matroids and the integral Hodge conjecture for abelian varieties

We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. Our proof is motivated by tropical geometry; it relies on multivariable Mumford constructions, monodromy considerations, and the combinatorial theory of matroids.

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arXiv2025

Cycle conjectures and birational invariants over finite fields

We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the Grothendieck--Serre semi-simplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1-semi-simplicity conjecture in half of the degrees implies those conjectures in all degrees.

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arXiv2025

On the rationality problem for low degree hypersurfaces

We show that a very general hypersurface of degree d at least 4 and dimension at most $(d+1)2^{d-4}$ over a field of characteristic different from 2 does not admit a decomposition of the diagonal; hence, it is neither stably nor retract rational, nor $\mathbb{A}^1$-connected. Similar results hold in characteristic 2 under a slightly weaker degree bound. This improves earlier results by the second named author and Moe.

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arXiv2025

Abelian varieties with no power isogenous to a Jacobian

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a product of Jacobians of curves. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties.

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arXiv2024

A moving lemma for cohomology with support

For a natural class of cohomology theories with support (including étale or pro-étale cohomology with suitable coefficients), we prove a moving lemma for cohomology classes with support on smooth quasi-projective k-varieties that admit a smooth projective compactification (e.g. if char(k)=0). This has the following consequences for such k-varieties and cohomology theories: a local and global generalization of the effacement theorem of Quillen, Bloch--Ogus, and Gabber, a finite level version of the Gersten conjecture in characteristic zero, and a generalization of the injectivity property and the codimension 1 purity theorem for étale cohomology. Our results imply that the refined unramified cohomology groups from [Sch23] are motivic.

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arXiv2024

Remarks on Milnor K-theory and Tate's conjecture for divisors

We show that the Tate conjecture for divisors over a finite field $\mathbb F$ is equivalent to an explicit algebraic problem about the third Milnor K-group of the function field $\bar {\mathbb F}(x,y,z)$ in three variables over $\bar {\mathbb F}$.

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arXiv2024

Truncated pushforwards and refined unramified cohomology

For a large class of cohomology theories, we prove that refined unramified cohomology is canonically isomorphic to the hypercohomology of a natural truncated complex of Zariski sheaves. This generalizes a classical result of Bloch and Ogus and solves a conjecture of Kok and Zhou.

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arXiv2023

Geometric retract rationality of norm varieties

Let k be a field of characteristic zero. We show that the norm variety associated to a prime $\ell$ and an ordered sequence of invertible elements of k is geometrically retract rational. This generalizes a recent result of Balwe-Hogadi-Sawant, where geometric $\mathbb A^1$-connectedness (an a priori weaker notion) had been proven.

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arXiv2023

The diagonal of quartic fivefolds

We show that a very general quartic hypersurface in $\mathbb P^6 $ over a field of characteristic different from 2 does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a result of Nicaise--Ottem, who showed stable irrationality over fields of characteristic 0. To prove our result, we introduce a new cycle-theoretic obstruction that may be seen as an analogue of the motivic obstruction for rationality in characteristic zero, introduced by Nicaise--Shinder and Kontsevich--Tschinkel.

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