John Lesieutre
Papers:

"Higher arithmetic degrees of dominant rational selfmaps", submitted. (arXiv:1906.11188)
abstract±
Suppose that f : X ⇢ X is a dominant rational selfmap of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if P ∈ Q is a point with welldefined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_{1}(f) of f if the orbit of P is Zariski dense in X.
In this note, we extend the KawaguchiSilverman conjecture to the setting of orbits of higherdimensional subvarieties of X. We begin by defining a set of arithmetic degrees of f, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

"Notions of numerical Iitaka dimension do not coincide", submitted. (arXiv:1904.10832)
abstract±
Let X be a smooth projective variety. The Iitaka dimension of a divisor D is an important invariant, but it does not only depend on the numerical class of D. However, there are several definitions of "numerical Iitaka dimension", depending only on the numerical class. In this note, we show that there exists a pseuodoeffective Rdivisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective Rdivisor D_{+} for which h^{0}(X,⌊mD_{+}⌋+A) is bounded above and below by multiples of m^{3/2} for any sufficiently ample A.

"A rational map with infinitely many points of distinct arithmetic degrees" (with M. Satriano), to appear in Ergodic Theory and Dynamical Systems. (arXiv:1809.00047)
abstract±
Let f : X ⇢ X be a dominant rational selfmap of a smooth projective variety defined over Q. For each point P ∈ Q whose forward forbit is welldefined, Silverman introduced the arithmetic degree α_{f}(P), which measures the growth rate of the heights of the points f^{n}(P). Kawaguchi and Silverman conjectured that α_{f}(P) is welldefined and that, as P varies, the set of values obtained by α_{f}(P) is finite. Based on constructions of BedfordKim and McMullen, we give a counterexample to this conjecture when X = P^{4}.

"Canonical heights on hyperKähler varieties and the Kawaguchi–Silverman conjecture" (with M. Satriano), to appear in Int. Math. Res. Not.. (arXiv:1802.07388)
abstract±
The Kawaguchi–Silverman conjecture predicts that if f : X ⇢ X is a dominant rationalself map of a projective variety over Q, and P is a Qpoint of X with Zariskidense orbit, then the dynamical and arithmetic degrees of f coincide: λ_{1}(f)=α_{f}(P). We prove this conjecture in several higherdimensional settings, including all endomorphisms of nonuniruled smooth projective threefolds with degree larger than 1, and all endomorphisms of hyperKähler varieties in any dimension. In the latter case, we construct a canonical height function associated to any automorphism f : X → X of a hyperKähler variety defined over Q.

"A projective variety with discrete, nonfinitely generated automorphism group", Inventiones Math 212 (2018), no. 1, 189–211. (arXiv:1609.06391) (sage, output)
abstract±
We construct a projective variety with discrete, nonfinitely generated automorphism group. As an application, we show that there exists a complex projective variety with infinitely many nonisomorphic real forms.

"Dynamical MordellLang and automorphisms of blowups" (with D. Litt), Algebraic Geometry 6 (2019), no. 1, 1–25. (arXiv:1604.08216)
abstract±
We show that if φ : X → X is an automorphism of a smooth projective variety and D ⊂ X is an irreducible divisor for which the set of d in D with φ^n(d) in D for some nonzero n is not Zariski dense, then (X, φ) admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of Aut(X), extending results of BayraktarCantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism.
These results follow from a nonreduced analogue of the dynamical MordellLang conjecture. Namely, let φ : X → X be an étale endomorphism of a smooth projective variety X over a field k of characteristic zero. We show that if Y and Z are two closed subschemes of X, then the set A_{φ}(Y,Z) = {n : φ^{n}(Y) ⊂ Z} is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of Y.

"Log Fano structures and Cox rings of blowups of products of projective spaces" (with J. Park), Proc. Amer. Math. Soc. 145 (2017), no. 10, 4201–4209. (arXiv:1604.07140)
abstract±
The aim of this paper is twofold. Firstly, we determine which blowups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Secondly, we describe generators of the Cox rings of some cases.

"Effective cones of cycles on blowups of projective space" (with I. Coskun and J.C. Ottem), Algebra & Number Theory 109 (2016). (arXiv:1603.04808)
abstract±
In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blowups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blowup X^{n}_{r} of P^{n}, n>4, at r very general points, the cone of divisors is not finitely generated as soon as r>n+3, whereas the cone of curves is generated by the classes of lines if r ≤ 2^{n}. In fact, if X^{n}_{r} is a Mori Dream Space then all the effective cones of cycles on X^{n}_{r} are finitely generated.

"A few questions about curves on surfaces" (with Ciliberto, Knutsen, Lozovanu, Miranda, Mustopa, and Testa), Rend. Circ. Mat. Palermo (2016), 1–10. (arXiv:1511.06618)
abstract±
In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and D a divisor on X satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on D or on X.

"Some constraints on positive entropy automorphisms of smooth threefolds", to appear in Ann. Sci. École Norm. Sup.. (arXiv:1503.07834)
abstract±
Suppose that X is a smooth, projective threefold over C and that φ : X → X is an automorphism of positive entropy. We show that one of the following must hold, after replacing φ by an iterate: i) the canonical class of X is numerically trivial; ii) φ is imprimitive; iii) φ is not dynamically minimal. As a consequence, we show that if a smooth threefold M does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blowups of M can admit a primitive automorphism of positive entropy. In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a nonuniruled, terminal threefold X with infinitely many K_{X}negative extremal rays on NE(X).

"A pathology of asymptotic multiplicity in the relative setting", Math. Res. Lett. 23 (2016), no. 5, 1433–1451. (arXiv:1502.03019)
abstract±
We point out an example of a projective family π : X → S, a πpseudoeffective divisor D on X, and a subvariety V ⊂ X for which the asymptotic multiplicity σ_{V}(D;X/S) is infinite. This shows that the divisorial Zariski decomposition is not always defined for pseudoeffective divisors in the relative setting.

"Curves disjoint from a nef divisor" (with J.C. Ottem), Michigan Math. J. 65 (2016), 321–332.(arXiv:1410.4467)
abstract±
On a projective surface it is wellknown that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing an effective, nef line bundle on a threefold which is trivial on countably infinitely many curves. This answers a question of Totaro. As a pleasant corollary, we exhibit a quasiprojective variety with only a countably infinite set of complete, positivedimensional subvarieties.

"Derivedequivalent rational threefolds", Int. Math. Res. Not. (2015) 6011–6020. (journal / arXiv:1311.0056)
abstract±
We describe an infinite set of smooth projective threefolds that have equivalent derived categories but are not isomorphic, contrary to a conjecture of Kawamata. These arise as blowups of P^{3} at various configurations of 8 points, which are related by Cremona transformations.

"The diminished base locus is not always closed", Compositio Math. 150 (2014), no. 10, 1729–1741. (journal / arXiv:1212.3738)
abstract±
We exhibit a pseudoeffective Rdivisor D_{∞} on the blowup of P^{3} at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus B_{}(D_{∞}) = ∪_{A ample} B(D_{∞}+A) is not closed and that D_{∞} does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an Rdivisor on the family of blowups of P^{2} at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.