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János Kollár

Publications and source records attributed to János Kollár.

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.

math.AG↗
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.

math.AG↗
arXiv2026

Sections of Jacobian fibrations over lines

Let $|H|$ be a linear system on a smooth surface $S$. We study the cohomology classes of sections of the universal Jacobian over lines in $|H|$. When $S$ is a K3 surface, the universal compactified Jacobian is a hyperkähler manifold, and we do not see how to reconcile our results with some of the steps in [BKV25]. v.2: Confusing typo fixed: H^g should be H^{2g}. v3: Note added: Conjecture 7 is now corrected and solved by Pascal Autissier and Andrea Fanelli (On a conjecture by Kollár and Saccà, arXiv:2606.26983).

math.AG↗
arXiv2026

Theorems of Bertini and Chevalley

We give a short proof of Chevalley's theorem that every algebraic group is an extension of an Abelian variety by a linear algebraic group. Along the way we treat Bertini's irreducibility theorem.

math.AG↗
arXiv2025

Higher direct images of dualizing sheaves III

We show that for flat morphisms between varieties with rational singularities, the higher direct images of the structure sheaf are locally free. As a consequence, the identity component of the relative Picard scheme is a smooth algebraic group scheme. v.2. Bhatt pointed out that Lemma 9 was incorrect; it is now replaced. The main theorem is unchanged.

math.AG↗
arXiv2025

Arthur Byron Coble, 1878--1966

A short essay on the life and mathematical heritage of Coble. A substantially edited version will be part of the series of biographical memoirs of past members of the National Academy of Sciences. Version 2: minor changes. Version 3. Typo corrected.

math.HO↗
arXiv2025

Severi-Brauer varieties; a geometric treatment

These notes present a geometric treatment of Severi-Brauer varieties, without using any results from the theory of central simple algebras or from Galois cohomology. 2026 version: major revisions

math.AG↗
arXiv2025

Essential dimension of isogenies

We give a lower bound for the essential dimension of isogenies of complex abelian varieties. The bound is sharp in many cases. In particular, the multiplication-by-$m$ map is incompressible for every $m\geq 2$, confirming a conjecture of Brosnan.

math.AG↗
arXiv2025

Néron models, minimal models, and birational group actions

Let $A_K$ be an Abelian variety over the quotient field of a Dedekind domain $R$. We show that the identity component of the Néron model of $A_K$ acts regularly on any minimal model of $A_K$ over $R$, and discuss when the Néron model is an open subset of a minimal model. The main technical result is the rigidity of small modifications, leading to a regularity criterion for birational group actions.

math.AG↗
arXiv2024

Cubic surfaces with infinite, discrete automorphism group

We prove that the automorphism group of an affine, cubic surface with equation $xyz=g(x,y)$ contains ${\mathbb Z}$ as a finite index subgroup. These equations were first studied by Mordell. v.2: small changes, references updated.

math.AG↗
arXiv2024

Log canonical singularities of plurisubharmonic functions

A plurisubharmonic weight is log canonical if it is at the critical point of turning non-integrable. Given a log canonical plurisubharmonic weight, we show that locally there always exists a log canonical `holomorphic' weight having the same non-integrable locus. The proof uses the theory of quasimonomial valuations developed by Jonsson-Mustaţǎ and Xu, as well as the minimal model program over complex analytic spaces due to Fujino and Lyu-Murayama. As a consequence, we obtain that the non-integrable locus is seminormal.

math.CV↗