arXiv subjects

Tomasz Pełka

Publications and source records attributed to Tomasz Pełka.

12 recordsLinked to original sources

arXiv2026

Fibrations by Lagrangian tori for maximal Calabi-Yau degenerations and beyond

We introduce a general technique to construct Lagrangian torus fibrations in degenerations of Kähler manifolds. We show that such torus fibrations naturally occur at the boundary of the A'Campo space. This space extends a degeneration over a punctured disc to a hybrid space involving its real oriented blowup and the dual complex of the special fiber; equipped with an exact modification of a given Kähler form which becomes fiberwise symplectic at the boundary. Therefore, it allows to move Lagrangian tori from the "radius zero" fibers at the boundary to the nearby fibers of the original generation using the symplectic connection. As a consequence, for any maximal Calabi-Yau degeneration we prove that a generic region of each nearby fiber admits a Lagrangian torus fibration with asymptotic properties expected from the (conjectural) Strominger--Yau--Zaslow fibration.

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arXiv2026

Saturation of algebraic surfaces

The saturation of an algebraic surface is the maximal open embedding with complement of dimension zero. For schemes, it was introduced by the first named author and A. Bondal who asked whether every saturated surface is proper over its affinisation. We prove that this property holds whenever the affinisation is non-trivial. If a saturated surface has trivial affinisation, we prove that the boundary of any compactification has at most two connected components, and this bound is optimal. Furthermore, we extend the results of A. Bondal and the first named author from schemes to algebraic spaces; in particular, we prove that the saturation of a surface can be recovered from the category of reflexive sheaves.

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arXiv2025

On the structure of open del Pezzo surfaces

Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show that, up to well described exceptions in characteristics 2, 3 and 5, the smooth part of $X\setminus D$ admits an $\mathbb{A}^1$- or an $\mathbb{A}^{1}_{*}$-fibration, which extends to a $\mathbb{P}^1$-fibration of the minimal log resolution of $(X,D)$. In characteristic 0 this improves a well-known structure theorem of Miyanishi-Tsunoda. Within the proof, we classify rational anti-canonical curves contained in smooth loci of canonical del Pezzo surfaces of rank one.

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arXiv2025

Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries

This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo surface, defined as the minimal intersection number of the exceptional divisor of the minimal resolution and a fiber of some $\mathbb{P}^1$-fibration. The geometry of del Pezzo surfaces gets more constrained as the height grows; in characteristic $0$ no example of height bigger than $4$ is known. In this article, we classify del Pezzo surfaces of Picard rank one and height at most $2$; in particular we describe the non-log terminal ones. We also describe a natural class of del Pezzo surfaces which have descendants with elliptic boundary, i.e. whose minimal resolution has a birational morphism onto a canonical del Pezzo surface of rank one mapping the exceptional divisor to an anti-canonical curve.

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arXiv2025

Classification of del {P}ezzo surfaces of rank one. III. Height 3

This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the height, defined as the minimal number $h$ such that the minimal resolution of singularities admits a $\mathbb{P}^1$-fibration whose fiber meets the exceptional divisor $h$ times. It is expected that every singular del Pezzo surface of rank one is of height $h\leq 4$, with minor exceptions in characteristics $2$ and $3$. Having settled the case $h\leq 2$ in our previous article arXiv:2412.21174, we now give a classification in case the height equals $3$.

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arXiv2024

On the embedded Nash problem

The embedded Nash problem for a hypersurface in a smooth algebraic variety, is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

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arXiv2024

Symplectic monodromy at radius zero and equimultiplicity of $μ$-constant families

We show that every family of isolated hypersurface singularity with constant Milnor number has constant multiplicity. To achieve this, we endow the A'Campo model of "radius zero" monodromy with a symplectic structure. This new approach allows to generalize a spectral sequence of McLean converging to fixed point Floer homology of iterates of the monodromy to a more general setting which is well suited to study $μ$-constant families.

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arXiv2023

Smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture

A complex algebraic surface $S$ is a $\mathbb{Q}$-homology plane if $H_{i}(S,\mathbb{Q})=0$ for $i>0$. The Negativity Conjecture of Palka asserts that $κ(K_{X}+\tfrac{1}{2}D)=-\infty$, where $(X,D)$ is a log smooth completion of $S$. We give a complete description of smooth $\mathbb{Q}$-homology planes satisfying the Negativity Conjecture. We restrict our attention to those of log general type, as otherwise their geometry is well understood. We show that, as conjectured by tom Dieck and Petrie, they can be arranged in finitely many discrete series, each obtained in a uniform way from an arrangement of lines and conics on $\mathbb{P}^{2}$. We infer that these surfaces satisfy the Rigidity Conjecture of Flenner and Zaidenberg; and a conjecture of Koras, which asserts that $\#\operatorname{Aut}(S)\leq 6$.

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arXiv2021

Linearization of holomorphic families of algebraic automorphisms of the affine plane

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.

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arXiv2021

Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units

We give a corrected statement of the theorem of Gurjar and Miyanishi, which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such surfaces by $\mathcal{S}_{0}$. An infinite series of surfaces in $\mathcal{S}_{0}$, not listed by Gurjar and Miyanishi, was recently obtained by Freudenburg, Kojima and Nagamine as affine modifications of the plane. We complete their list to a series containing arbitrarily high-dimensional families of pairwise non-isomorphic surfaces in $\mathcal{S}_{0}$. Moreover, we classify them up to a diffeomorphism, showing that each occurs as an interior of a 4-manifold whose boundary is an exceptional surgery on a 2-bridge knot. In particular, we show that $\mathcal{S}_{0}$ contains countably many pairwise non-homeomorphic surfaces.

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arXiv2019

Classification of planar rational cuspidal curves. II. Log del Pezzo models

Let $E\subseteq \mathbb{P}^2$ be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira-Iitaka dimension of $K_X+\frac{1}{2}D$, where $(X,D)\to (\mathbb{P}^{2},E)$ is a minimal log resolution, is negative. We prove structure theorems for curves satisfying this conjecture and we finish their classification up to a projective equivalence by describing the ones whose complement admits no $\mathbb{C}^{**}$-fibration. As a consequence, we show that they satisfy the Strong Rigidity Conjecture of Flenner-Zaidenberg. The proofs are based on the almost minimal model program. The obtained list contains one new series of bicuspidal curves.

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arXiv2016

Classification of planar rational cuspidal curves. I. C**-fibrations

To classify complex rational cuspidal curves $E\subseteq \mathbb{P}^2$ it remains to classify the ones with complement of log general type, i.e. the ones for which $κ(K_X+D)=2$, where $(X,D)$ is a log resolution of $(\mathbb{P}^2,E)$. It is conjectured that $κ(K_X+\frac{1}{2}D)=-\infty$ and hence $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibered, where $\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$, or $-(K_X+\frac{1}{2}D)$ is ample on some minimal model of $(X,\frac{1}{2}D)$. Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is $\mathbb{C}^{**}$-fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.

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