Abelian Threefolds Research Articles

On the integral Hodge conjecture for real abelian threefolds

Abstract We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds whose real locus is connected, and for real abelian threefolds A which are the product A = B × E {A=B\times E} of an abelian surface B and an elliptic curve E with connected real locus E ⁢ ( ℝ ) {E(\mathbb{R})} . Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the principally polarized case to the Jacobian case.

Lifts of Supersingular Abelian Varieties With Small Mumford–Tate Groups

Abstract We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford–Tate group. We prove that supersingular abelian surfaces, respectively three-folds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian three-folds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.

Basepoint-freeness thresholds and higher syzygies of abelian threefolds

For a polarized abelian variety, Z. Jiang and G. Pareschi introduce an invariant and show that the polarization is basepoint free or projectively normal if the invariant is small. Their result is generalized to higher syzygies by F. Caucci, that is, the polarization satisfies property $(N_p)$ if the invariant is small. In this paper, we study a relation between the invariant and degrees of abelian subvarieties with respect to the polarization. For abelian threefolds, we give an upper bound of the invariant using degrees of abelian subvarieties. In particular, we affirmatively answer some questions on abelian varieties asked by the author, V. Lozovanu and Caucci in the three dimensional case.

A classification of the automorphism groups of polarized abelian threefolds over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.

Spanning the isogeny class of a power of an elliptic curve

Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions

We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.

Mass formula and Oort's conjecture for supersingular abelian threefolds

Katsura and Oort obtained an explicit description of the supersingular locus S3,1 of the Siegel modular variety of degree 3 in terms of class numbers. In this paper we study an alternative stratification of S3,1, the so-called mass stratification. We show that when p≠2, there are eleven strata (one of a-number 3, two of a-number 2 and eight of a-number 1). We give an explicit mass formula for each stratum and classify possible automorphism groups on each stratum of a-number one. On the largest open stratum we show that every automorphism group is {±1} if and only if p≠2; that is, we prove that Oort's conjecture on the automorphism groups of generic supersingular abelian threefolds holds precisely when p>2.

Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate-Shafarevich groups. In particular, when the modular curve $X_0(3p)$ has infinitely many $F$-rational points the method applies to ``most'' elliptic curves $E$ having a cyclic $3p$-isogeny. It also applies in certain cases when $X_0(3p)$ has only finitely many points. For example, we find an elliptic curve over $\mathbb{Q}$ for which a positive proportion of quadratic twists have an element of order $5$ in their Tate-Shafarevich groups. The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime $p \equiv 1 \pmod 9$, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate-Shafarevich groups.

Derived categories and the genus of space curves

We generalize a classical result about the genus of curves in projective space by Gruson and Peskine to principally polarized abelian threefolds of Picard rank one. The proof is based on wall-crossing techniques for ideal sheaves of curves in the derived category. In the process, we obtain bounds for Chern characters of other stable objects such as rank two sheaves. The argument gives a proof for projective space as well. In this case these techniques also indicate an approach for a conjecture by Hartshorne and Hirschowitz and we prove first steps towards it.

Groups of points on abelian threefolds over finite fields

In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given Fq-isogeny class. This work completes partial classification given in [9].

Reduced Donaldson-Thomas invariants and the ring of dual numbers

Let $A$ be an abelian variety. We introduce $A$-equivariant Grothendieck rings and $A$-equivariant motivic Hall algebras, and endow them with natural integration maps to the ring of dual numbers. The construction allows a systematic treatment of reduced Donaldson-Thomas invariants by Hall algebra techniques. We calculate reduced Donaldson-Thomas invariants for $\mathrm{K3} \times E$ and abelian threefolds for several imprimitive curve classes. This verifies (in special cases) multiple cover formulas conjectured by Oberdieck-Pandharipande and Bryan-Oberdieck-Pandharipande-Yin.

Curve counting on abelian surfaces and threefolds

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by Gottsche-Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the generating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, Goettsche, and Lange-Sernesi. Further evidence is provided by Donaldson-Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject.

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps: Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property.

Residual representations of semistable principally polarized abelian varieties

Let $A/{\mathbb {Q}}$ be a semistable principally polarized abelian variety of dimension d≥1. Let ℓ be a prime and let $\overline {\rho }_{A,\ell }\colon G_{\mathbb {Q}} \rightarrow \text {GSp}_{2d}({\mathbb {F}}_{\ell })$ be the representation giving the action of $G_{\mathbb {Q}} :=\text {Gal}(\overline {{\mathbb {Q}}}/{\mathbb {Q}})$ on the ℓ-torsion group A[ℓ]. We show that if ℓ≥ max(5,d+2), and if image of $\overline {\rho }_{A,\ell }$ contains a transvection then $\overline {\rho }_{A,\ell }$ is either reducible or surjective. With the help of this we study surjectivity of $\overline {\rho }_{A,\ell }$ for semistable polarized abelian threefolds, and give an example of a genus 3 hyperelliptic curve $C/{\mathbb {Q}}$ such that $\overline {\rho }_{J,\ell }$ is surjective for all primes ℓ≥3, where J is the Jacobian of C.

Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds II

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian threefold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov–Gieseker (BG) type inequality. This is done by showing certain Fourier–Mukai transforms (FMTs) give equivalences of abelian categories which are double tilts of coherent sheaves.

Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds

We use the ideas of Bayer, Bertram, Macr and Toda to construct a Bridgeland stability condition on a principally polarized abelian threefold (X;L) with NS(X) = Z[‘] by establishing their Bogomolov{Gieseker-type inequality for certain tilt stable objects associated with the pair (A p 3‘=2;‘=2 ;Z p 3‘=2;‘=2 ) on X. This is done by proving the stronger result thatA p 3‘=2;‘=2 is preserved by a suitable Fourier{Mukai transform.

Vector Bundles and Monads On Abelian Threefolds

The purpose of this article is to construct examples of stable rank 2 vector bundles on abelian threefolds and to study their moduli. More precisely, we consider principally polarized abelian threefolds (X, Θ) with Picard number 1. Using the Serre construction, we obtain stable rank 2 bundles realizing roughly one half of the Chern classes (c 1, c 2) that are a priori allowed by the Bogomolov inequality and Riemann–Roch. In the case of even c 1, we study first order deformations of these vector bundles ℰ, using a second description in terms of monads, similar to the ones used by Barth–Hulek on projective space. We find that all first order deformations of the bundle are induced by first order deformations of the corresponding monad, which leads to the formula where Δ denotes the discriminant . In the simplest nontrivial case (where c 1 = 0 and c 2 = Θ2), we construct an explicit parametrization of a Zariski open neighbourhood of ℰ in its moduli space: this neighbourhood is a ruled, nonsingular variety of dimension 13, birational to a ℙ1-bundle over X × X × H, where H is the Hilbert scheme (of Kodaira dimension zero) of two points on the Kummer threefold X/(−1).

Siegel modular forms of degree three and the cohomology of local systems

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space $$\mathcal{A }_3$$ of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of degree three, such as dimension formulas and traces of Hecke operators. We also use it to predict several lifts from genus 1 to genus 3, as well as lifts from $$\mathrm{G }_2$$ and new congruences of Harder type.

Donaldson–Thomas invariants for complexes on abelian threefolds

We modify the standard perfect symmetric obstruction theory for moduli spaces of simple perfect complexes, to the situation of complexes on abelian threefolds with fixed determinant and Fourier-Mukai determinant. As outcome we attach nontrivial Donaldson-Thomas invariants to moduli spaces for complexes modulo twist and translation.

Jacobians among abelian threefolds: a geometric approach

Let (A, θ) be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over \({\bar k}\) . There exists a canonical extension k′/k, of degree ≤ 2, such that (A, θ) becomes isomorphic to a Jacobian over k′. The aim of this note is to give a geometric construction of this extension.