Kodaira-Spencer rank distribution in Buium's delta-stratification of the moduli of principally polarized abelian varieties: simple abelian varieties, Jacobians, and Manin-kernel absolute dimensions

Around Buium's delta-stratification of the moduli space of principally polarized abelian varieties - the loci of abelian varieties whose differential-algebraic Kodaira-Spencer (KS) rank is at most a given bound - Dupuy and Freitag asked two questions that govern the supply of strongly minimal order-one differential equations: for each dimension g and each 1 <= j <= g, does the stratum W_{j,g,n} \ W_{j-1,g,n} contain a simple abelian variety, and does it meet the Torelli locus of Jacobians of curves? Because the absolute dimension of the Manin kernel of a simple abelian variety A of dimension g equals g plus its KS-rank, the first question is equivalent to asking which Manin-kernel absolute dimensions occur for simple A; the source authors conjecture that every integer h >= 3 occurs as the Manin-kernel absolute dimension of a nonisotrivial simple abelian variety, announced a nonconstructive proof over differentially closed fields in joint work with Scanlon (unpublished), and leave constructive realizations over finitely generated differential fields open. The source realizes intermediate KS-ranks for simple abelian varieties in every odd dimension and for Jacobians in genera 3, 5, and 7 via CM families; the Coleman-Oort conjecture would confine that method to genus at most 7. The distribution questions remain open: no resolution is known in the literature, and the announced nonconstructive proof is not publicly available.

Background

The model theory of differentially closed fields of characteristic zero, denoted $\mathrm{DCF}_0$, studies algebraic ordinary differential equations through definability. A differential field $(K,\delta)$ carries a derivation, and in a model of $\mathrm{DCF}_0$ the constants $C=\{x:\delta x=0\}$ form an algebraically closed field. A definable set is strongly minimal if every definable subset of it is finite or cofinite; strongly minimal sets are the atoms of geometric stability theory, and the Zilber trichotomy divides them into geometrically trivial sets (any two generic elements are independent), modular nontrivial sets (group-like), and nonmodular sets. This trichotomy is a workhorse in applications of the model theory of differential fields, so the inventory of strongly minimal sets of each kind is basic knowledge for the subject.

Two families of examples organize the question. First, to an abelian variety $A$ of dimension $g$ over $(K,\delta)$ with algebraically closed constants one attaches its Manin kernel $A^{\sharp}$, the differential algebraic subgroup of $A$ obtained as the Kolchin closure (closure in the differential-Zariski topology, whose closed sets are zero loci of differential polynomial equations) of the torsion of $A$; Manin kernels played a central role in the differential-algebraic approach to Mordell-Lang-type statements (describing the points of a subvariety of an algebraic group that lie in a translate of a subgroup). The absolute dimension $a(A^{\sharp})$ (the order of the defining differential equations, equivalently the transcendence degree of a generic point) satisfies $g\le a(A^{\sharp})\le 2g$, with the lower bound attained exactly when $A$ is isotrivial, i.e., descends to the constants. For simple $A$ (no proper abelian subvarieties) the Manin kernel $A^{\sharp}$ is strongly minimal, and the Manin kernels of simple abelian varieties exhaust the nonorthogonality classes of modular nontrivial strongly minimal sets in $\mathrm{DCF}_0$ (two strongly minimal sets are nonorthogonal when they are not mutually generic-independent, and this equivalence relation partitions them into classes); hence knowing exactly which absolute dimensions occur is direct input for the trichotomy. For instance, if no nonisotrivial abelian surface had a Manin kernel of absolute dimension $3$, then every strongly minimal set of absolute dimension $3$ would automatically be geometrically trivial - a typical use of such dimension restrictions. Before the source paper, the only known simple abelian variety with intermediate dimension $g<a(A^{\sharp})<2g$ was an abelian fourfold obtained indirectly by Bertrand and Pillay.

Second, order-one differential equations on algebraic curves. Hrushovski and Itai gave a detailed theory of autonomous (constant-coefficient) order-one equations on curves via pairs $(C,\omega)$ of a curve and a regular differential form, showing that 'essential' forms yield geometrically trivial strongly minimal sets; Freitag and Moosa later proved that no order-one equation has the then-sought property of being geometrically trivial and strongly minimal with infinitely many algebraic solutions, so new examples of that shape had to come from non-autonomous equations. Rosen extended the pairs $(C,\omega)$ formalism to curves not defined over the constants: the derivative of a point of a variety $X/K$ no longer lies in the tangent bundle but in a torsor $\tau X$ over it (the first prolongation, or jet space), whose cohomology class in $H^{1}(X,T_{X/K})$ is the Kodaira-Spencer class measuring how $X$ varies under $\delta$. The dual object to $\tau X$ is the sheaf $\Omega^{\tau}_{X/K}$ of KS-forms, and order-one equations on a non-autonomous curve are governed by its global sections. Dupuy and Freitag developed this theory and answered Rosen's question, constructing strictly disintegrated (a strong form of geometrically trivial: any two generic solutions are independent) strongly minimal order-one equations on nonisotrivial curves.

The computational backbone is Buium's differential-algebraic Kodaira-Spencer rank: for a smooth variety $X/K$ it is the rank of the cup-product-with-Kodaira-Spencer map $H^{0}(X,\Omega^{1}_{X/K})\to H^{1}(X,\mathcal{O}_{X})$, $\omega\mapsto\omega\cup\mathrm{KS}_{X/K}(\delta)$. For $g\ge 2$ and $n\ge 3$, Buium proved that the loci $\mathcal{W}_{j,g,n}\subset\mathcal{A}_{g,n}(K)$ of principally polarized abelian varieties of dimension $g$ (abelian varieties, i.e., projective commutative algebraic groups, equipped with a principal polarization) with level-$n$ structure (a compatible choice of symplectic basis of the $n$-torsion) whose KS-rank is at most $j$ form a stratification $\mathcal{W}_{0,g,n}\subset\mathcal{W}_{1,g,n}\subset\cdots\subset\mathcal{W}_{g,g,n}=\mathcal{A}_{g,n}(K)$ by delta-subvarieties cut out by order-one equations, all nonempty - nonemptiness being achieved with products of elliptic curves, which are not simple - while a Zariski-generic point has maximal rank. Dupuy and Freitag prove the identities $a(A^{\sharp})=g+\operatorname{rk}(\mathrm{KS}_{A/K}(\delta))$ and $h^{0}(C,\Omega^{\tau}_{C/K})=2g+1-a(\operatorname{Jac}C^{\sharp})$, so the KS-rank spectrum, the set of achievable Manin-kernel absolute dimensions, and the existence of 'new' KS-forms on curves (forms not pulled back from lower genus, which produce strictly disintegrated order-one equations because the old forms occupy a countable union of proper subspaces once $h^{0}\ge 2$) are three readings of one quantity. Using Andre's decomposition of the cup-product-with-Kodaira-Spencer map for abelian schemes with CM (complex multiplication, i.e., with endomorphism algebra a CM number field - a totally imaginary quadratic extension of a totally real number field), they realize simple abelian varieties with intermediate KS-rank in every odd dimension and simple abelian varieties with KS-rank between 1 and 2 in every dimension; via families of CM Jacobians (Picard curves, the cyclic triple covers $y^{3}=f(x)$ with $\deg f=4$, of genus 3 going back to Koike-Weng, and the prime-genus families of de Jong-Noot, Moonen, and Frediani-Ghigi-Penegini in genera 5 and 7) they realize nonisotrivial curves whose simple Jacobians have intermediate KS-rank. Arithmetic bounds the method: the Coleman-Oort conjecture predicts that no positive-dimensional families of simple CM Jacobians exist for $g\ge 8$ (Moonen reduced it to three key cases), so the CM route cannot reach large genus. What remains open is the distribution itself: whether every stratum is populated by simple abelian varieties and by Jacobians, and whether the full spectrum of Manin-kernel dimensions can be realized constructively.

Problem Statement

Let $(K,\delta)$ be a differential field of characteristic zero with algebraically closed constants (for instance a model of $\mathrm{DCF}_0$), let $g\ge 2$ and $n\ge 3$, and let $\mathcal{A}_{g,n}$ be the moduli space of principally polarized abelian varieties of dimension $g$ with level-$n$ structure (abelian varieties, i.e., projective commutative algebraic groups, equipped with a principal polarization and a compatible choice of symplectic basis of their $n$-torsion). For an abelian variety $A/K$ let $\operatorname{rk}(\mathrm{KS}_{A/K}(\delta))$ denote Buium's differential-algebraic Kodaira-Spencer rank, i.e., the rank of the cup-product-with-Kodaira-Spencer map $H^{0}(A,\Omega^{1}_{A/K})\to H^{1}(A,\mathcal{O}_{A})$, $\omega\mapsto\omega\cup\mathrm{KS}_{A/K}(\delta)$, where $\mathrm{KS}_{A/K}(\delta)\in H^{1}(A,T_{A/K})$ is the Kodaira-Spencer class of $A$ over $(K,\delta)$, and let $\mathcal{W}_{j,g,n}\subset\mathcal{A}_{g,n}(K)$ be the delta-subvariety of points of rank at most $j$, so that $\mathcal{W}_{0,g,n}\subset\cdots\subset\mathcal{W}_{g,g,n}=\mathcal{A}_{g,n}(K)$ is Buium's stratification. Call $A$ simple if it has no abelian subvariety other than $0$ and itself. For simple $A$ of dimension $g$ the Manin kernel $A^{\sharp}$ - the differential algebraic subgroup of $A$ given as the Kolchin closure (differential-Zariski closure) of the torsion of $A$ - has absolute dimension $a(A^{\sharp})=g+\operatorname{rk}(\mathrm{KS}_{A/K}(\delta))$, where the absolute dimension is the transcendence degree of a generic point, equivalently the order of the defining differential equations.

Determine the distribution of Kodaira-Spencer ranks among simple abelian varieties and Jacobians, i.e. answer the following three linked questions, whose common objective is the achievable spectrum of ranks (equivalently, of Manin-kernel absolute dimensions) that controls the supply of strongly minimal order-one differential equations:

(a) For each $g\ge 2$ and each $j$ with $1\le j\le g$, does the stratum $\mathcal{W}_{j,g,n}\setminus\mathcal{W}_{j-1,g,n}$ contain a simple abelian variety? Equivalently, is every integer from $g+1$ to $2g$ attained as $a(A^{\sharp})$ for a simple abelian variety $A$ of dimension $g$; in particular, is every integer $h\ge 3$ attained as $a(A^{\sharp})$ for some nonisotrivial simple $A$ (the source authors' conjecture)?

(b) For each $g\ge 2$ and each $j$ with $1\le j\le g$, does $\mathcal{W}_{j,g,n}\setminus\mathcal{W}_{j-1,g,n}$ intersect the Torelli locus, i.e., the image in $\mathcal{A}_{g,n}$ of the moduli of smooth projective curves of genus $g$ under $C\mapsto\mathrm{Jac}(C)$ - equivalently, contain the Jacobian of a smooth projective curve of genus $g$ over $K$? This includes the large-genus regime (arbitrarily large $g$, beyond the CM-reachable range $g\le 7$): positive answers for $j\le g-1$ yield, by the source's criteria, global new KS-forms (global sections of the sheaf $\Omega^{\tau}_{C/K}$ of KS-forms, dual to the first prolongation torsor $\tau C$, that are not pulled back from lower-genus curves) and hence strictly disintegrated (a strong form of geometrically trivial: any two generic solutions are independent) strongly minimal order-one differential equations on nonisotrivial curves of genus $g$.

(c) Where existence holds, realize it constructively: for every integer $h\ge 3$, produce an explicit nonisotrivial simple abelian variety (equivalently, one with $\operatorname{rk}(\mathrm{KS}_{A/K}(\delta))\ge 1$, i.e. $g<a(A^{\sharp})\le 2g$) over a specified finitely generated differential field whose Manin kernel has absolute dimension $h$ (the source's CM-based constructions cover odd dimensions and ranks at most 2; a nonconstructive existence proof over differentially closed fields has been announced by the source authors with Scanlon but is unpublished, and constructive realizations over finitely generated fields, including uniform treatment of even dimensions, remain open).

The problem is completely answered when the set of realizable triples (dimension $g$, rank $j$, simple abelian variety or Jacobian) is determined - each stratum either provably contains such a point, with an explicit construction or existence proof, or provably does not - and the constructive realization in (c) is settled for every $h\ge 3$.

References

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