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Philip Engel

Publications and source records attributed to Philip Engel.

At least 19 recordsLinked to original sources
arXiv2026

On the degree of subvarieties on abelian varieties

Let $(X,\Theta)$ be a very general principally polarized abelian variety of dimension $g$, and consider the minimal cohomology class $\theta_k=[\Theta]^k/k!$ for $k<g$. We show that the minimal positive multiple of $\theta_k$ 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

Polync varieties and multiparameter Kulikov models

We study "polync varieties", whose singularities are locally products of normal crossing (nc) singularities. We introduce the notion of d-semistability of such varieties, and generalize work of Friedman and Kawamata-Namikawa to address the smoothability of d-semistable, K-trivial, polync varieties. These results are applications of recent breakthroughs on the logarithmic Bogomolov-Tian-Todorov theorem, due to Chan-Leung-Ma and Felten-Filip-Ruddat. We generalize the combinatorial description of Kulikov models for K3 surfaces to the setting of a multiparameter base and describe some interesting examples.

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arXiv2025

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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arXiv2025

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

Boundedness of some fibered K-trivial varieties

We prove that irreducible Calabi-Yau varieties of a fixed dimension, admitting a fibration by abelian varieties or primitive symplectic varieties of a fixed analytic deformation class, are birationally bounded. We prove that there are only finitely many deformation classes of primitive symplectic varieties of a fixed dimension, admitting a Lagrangian fibration. We also show that fibered Calabi-Yau 3-folds are bounded. Conditional on the generalized abundance or hyperk\"ahler SYZ conjecture, our results prove that there are only finitely many deformation classes of hyperk\"ahler varieties, of a fixed dimension, with $b_2 \geq 5$.

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arXiv2025

On lattice-polarized K3 surfaces

We propose modifications to the commonly used definitions of lattice-polarized and lattice-quasipolarized smooth K3 surfaces, collecting various versions of the definition, and determining the effects of these choices on the resulting moduli space. We fill a gap in the theory, by replacing Weyl chambers with the new notion of a ``small cone'': the true datum in the definition of lattice quasipolarized K3 surfaces. In addition, we describe the separated moduli stack and moduli space for lattice-polarized K3 surfaces with $ADE$ singularities, an important notion for applications.

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arXiv2024

Shafarevich's conjecture for families of hypersurfaces over function fields

Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in $\mathbb{P}_{\mathbb C}^{n+1}$. We prove that the finiteness is uniform in $\mathcal{S}$ and we give examples where the result is sharp. We also prove similar results for certain complete intersections in $\mathbb{P}_{\mathbb C}^{n+1}$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem.

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arXiv2023

On the non-abelian Hodge locus I

We partially resolve conjectures of Deligne and Simpson concerning $\mathbb{Z}$-local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with $\mathbb{Q}$-anisotropic monodromy, we prove (1) a relative form of Deligne's finiteness theorem, for any family of quasi-projective varieties, and (2) algebraicity of the corresponding non-abelian Hodge locus.

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arXiv2023

Exact enumeration of fullerenes

A fullerene, or buckyball, is a trivalent graph on the sphere with only pentagonal and hexagonal faces. Building on ideas of Thurston, we use modular forms to give an exact formula for the number of oriented fullerenes with a given number of vertices.

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arXiv2023

Mixed mock modularity of special divisors

We prove that the generating series of special divisors in toroidal compactifications of orthogonal Shimura varieties is a mixed mock modular form. More precisely, we find an explicit completion using theta series associated to rays in the cone decomposition. The proof relies on intersection theory at the boundary of the Shimura variety.

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arXiv2022

Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution

There are $75$ moduli spaces $F_S$ of K3 surfaces with a nonsymplectic involution. We give detailed descriptions of Kulikov models for one-parameter degenerations in $F_S$. In the $50$ cases where the fixed locus of the involution has a component $C_g$ of genus $g\ge2$, we identify normalizations of the KSBA compactifications of $F_S$ via stable pairs $(X,\epsilon C_g)$, with explicit semitoroidal compactifications of $F_S$.

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arXiv2022

The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces

Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surface associated to $A$ has infinitely many $k$-points not contained in a rational curve arising from a genus $2$ curve in $A$, again in contrast to the situation over the algebraic closure of a finite field.

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arXiv2021

Compact moduli of K3 surfaces with a nonsymplectic automorphism

We construct a modular compactification via stable slc pairs for the moduli spaces of K3 surfaces with a nonsymplectic group of automorphisms under the assumption that some combination of the fixed loci of automorphisms defines an effective big divisor, and prove that it is semitoroidal.

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arXiv2021

The flex divisor of a K3 surface

The flex divisor of a primitively polarized K3 surface $(X,L)$ of degree $L^2=2d$ is, generically, the locus of all points $x\in X$ for which there exists a pencil $V\subset |L|$ whose base locus is $\{x\}$. We show that the flex divisor lies in the linear system $|n_dL|$ where $n_d=(2d+1)C(d)^2$ and $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $F_{2d}$ of polarized K3 surfaces.

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arXiv2021

Compact moduli of K3 surfaces

We construct geometric compactifications of the moduli space $F_{2d}$ of polarized K3 surfaces, in any degree $2d$. Our construction is via KSBA theory, by considering canonical choices of divisor $R\in |nL|$ on each polarized K3 surface $(X,L)\in F_{2d}$. The main new notion is that of a recognizable divisor $R$, a choice which can be consistently extended to all central fibers of Kulikov models. We prove that any choice of recognizable divisor leads to a semitoroidal compactification of the period space, at least up to normalization. Finally, we prove that the rational curve divisor is recognizable for all degrees.

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