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Simion Filip

Publications and source records attributed to Simion Filip.

At least 19 recordsLinked to original sources
arXiv2026

Universal affine bundles for compact complex manifolds

We construct a universal affine bundle $E$ over a compact complex manifold $X$ equipped with a probability measure $\mu$, which is affine over the vector space $H_{1,1}(X;\mathbb{R})$, and satisfies a number of natural properties. The bundle $E$ carries a natural action of the automorphism group of $(X,\mu)$, and provides potentials for all $dd^c$-closed $(1,1)$-forms on $X$. We relate our construction to lifts of the action of the automorphism group of a Calabi--Yau manifold to its universal torsor.

math.DS↗
arXiv2026

Nonlinearizable embeddings of elliptic curves in rational surfaces

We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_\delta$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.

math.DS↗
arXiv2025

Measure Rigidity beyond Homogeneous Dynamics

We describe recent work that extends some of the measure and topological rigidity results in dynamical systems from situations homogeneous under a Lie group to quite general manifolds.

math.DS↗
arXiv2025

Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method

We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theorems are derived, applies also to the case of a single diffeomorphism or $1$-parameter flow and establishes extra invariance of a class of measures that we call ``generalized u-Gibbs states''.

math.DS↗
arXiv2024

Finiteness of totally geodesic hypersurfaces

We prove that a closed negatively curved analytic Riemannian manifold that contains infinitely many totally geodesic hypersurfaces is isometric to an arithmetic hyperbolic manifold. Equivalently, any closed analytic Riemannian manifold with negative sectional curvature has only finitely many totally geodesic hypersurfaces, unless it has constant curvature.

math.DG↗
arXiv2023

The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds

We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi-Yau complete intersections known as Wehler N-folds. We find that the volume function exhibits a pathological behavior when N>=3, we obtain examples of a pseudoeffective R-divisor D for which the volume of D+sA, with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h^0(X,[mD]+A) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold.

math.AG↗
arXiv2023

Gaps in the support of canonical currents on projective K3 surfaces

We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing self-intersection. The only previously known such examples were due to McMullen on non-projective K3 surfaces and were constructed using positive entropy automorphisms with a Siegel disk. Our construction is based on a Zassenhaus-type estimate for commutators of automorphisms.

math.DS↗
arXiv2021

Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents

We develop a class of uniformizations for certain weight 3 variations of Hodge structure (VHS). The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, M\"oller, and Zorich on Lyapunov exponents. Additionally, we prove that the monodromy representations are log-Anosov, a dynamical property that has a number of global consequences for the VHS. We establish a strong Torelli theorem for the VHS and describe appropriate domains of discontinuity. Additionally, we classify the hypergeometric differential equations that satisfy our assumptions. We obtain several multi-parameter families of equations, which include the mirror quintic as well as the six other thin cases of Doran--Morgan and Brav--Thomas.

math.AG↗
arXiv2021

A cyclotomic family of thin hypergeometric monodromy groups in ${Sp}_4(\mathbb{R})$

We exhibit an infinite family of discrete subgroups of ${Sp}_4(\mathbb R)$ which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters $\left(\tfrac{N-3}{2N},\tfrac{N-1}{2N}, \tfrac{N+1}{2N}, \tfrac{N+3}{2N}\right)$ at infinity and maximal unipotent monodromy at zero, for any integer $N\geq 4$. Additionally, we relate the cones used for ping-pong in $\mathbb R^4$ with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian.

math.AG↗
arXiv2021

Canonical currents and heights for K3 surfaces

We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.

math.DS↗
arXiv2020

Asymptotic shifting numbers in triangulated categories

We introduce invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category. The invariants are analogous to Poincare translation numbers that are widely used in dynamical systems. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Additionally, we relate our shifting numbers to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich.

math.AG↗
arXiv2019

On pseudo-Anosov autoequivalences

Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the exponential growth rate of masses under iterates of the autoequivalence, and only depends on the choice of a connected component of the stability manifold. We then propose a new definition of pseudo-Anosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudo-Anosov autoequivalences on the derived categories of quintic Calabi-Yau threefolds and quiver Calabi-Yau categories. Finally, we prove that certain pseudo-Anosov autoequivalences on quiver 3-Calabi-Yau categories act hyperbolically on the space of Bridgeland stability conditions.

math.AG↗
arXiv2019

Tropical Dynamics of Area-Preserving Maps

We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.

math.DS↗
arXiv2018

Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics

We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case.

math.DS↗
arXiv2017

Notes on the Multiplicative Ergodic Theorem

The Oseledets Multiplicative Ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures at summer schools in Brazil, France, and Russia.

math.DS↗
arXiv2017

Smooth and Rough Positive Currents

We study the different notions of semipositivity for (1,1) cohomology classes on K3 surfaces. We first show that every big and nef class (and every nef and rational class) is semiample, and in particular it contains a smooth semipositive representative. By contrast, we show that there exist irrational nef classes with no closed positive current representative which is smooth outside a proper analytic subset. We use this to answer negatively two questions of the second-named author. Using a result of Cantat and Dupont, we also construct examples of projective K3 surfaces with a nef R-divisor which is not semipositive.

math.CV↗