Universally irreducible subvarieties of Siegel moduli spaces

Gabriele Mondello; Riccardo Salvati Manni

doi:10.1515/crelle-2023-0078

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Universally irreducible subvarieties of Siegel moduli spaces

Gabriele Mondello and Riccardo Salvati Manni

Published/Copyright: November 20, 2023

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 806

Abstract

A subvariety of a quasi-projective complex variety X is called “universally irreducible” if its preimage inside the universal cover of X is irreducible. In this paper we investigate sufficient conditions for universal irreducibility. We consider in detail complete intersection subvarieties of small codimension inside Siegel moduli spaces of any finite level. Moreover, we show that, for g ≥ 3 , every Siegel modular form is the product of finitely many irreducible analytic functions on the Siegel upper half-space ℍ g . We also discuss the special case of singular theta series of weight 1 2 and of Schottky forms.

Funding statement: The first-named author was partially supported by GNSAGA research group. Both authors were partially supported by PRIN 2022 grant “Moduli and Lie theory”.

A Subvarieties of ℳ g , Dehn twists and transvections

We recall that the mapping class group MCG g can be identified to the orbifold fundamental group of ℳ g . Moreover, the homomorphism induced by the Torelli morphism

t g : ℳ g → 𝒜 g

at the level of fundamental groups π 1 ⁢ ( t g ) : π 1 ⁢ ( ℳ g ) → π 1 ⁢ ( 𝒜 g ) identifies to the standard symplectic representation ρ : MCG g → Γ g = Sp 2 ⁢ g ⁡ ( ℤ ) . We recall the following classical result (see, for instance, [16, Theorem 6.4]).

Fact A.1.

The standard symplectic representation ρ is surjective.

The following is an immediate consequence.

Corollary A.2.

For every subgroup Γ of Γ g , the natural homomorphism

π 1 ⁢ ( 𝒥 g ⁢ ( Γ ) ) → Γ

is surjective, where J g ⁢ ( Γ ) is the Jacobian locus in A g ⁢ ( Γ ) .

We also recall that the Torelli morphism t g can be extended to a map t ¯ g : ℳ ¯ g → 𝒜 ¯ g from the Deligne–Mumford compactification ℳ ¯ g to the Satake compactification 𝒜 ¯ g .

In order to state Proposition A.3, we denote by δ h irr the locally-closed locus inside ℳ ¯ g that parametrizes curves whose normalization has genus g - h (the genus of a disconnected curve being the sum of the genera of its connected components): in other words, points in δ h irr are isomorphism classes of stable curves with exactly h non-disconnecting nodes (and possibly other disconnecting nodes). We observe that δ h irr is a subvariety of codimension h.

In this appendix we want to show the following result (recall Definition 8.1).

Proposition A.3 (Finding a pair of commuting transvections I).

Let Y be an irreducible subvariety in M g . Suppose that

Y ¯ intersects the locally-closed boundary stratum δ h irr for some h > 0 ,
the intersection Y ¯ ∩ δ h irr is non-compact.

Then the image of π 1 ⁢ ( Y sm ) → Γ g contains two linearly independent, commuting transvections.

In order to prove Proposition A.3, we first recall that ℳ ¯ g supports a universal family of stable curves with total space 𝒞 ¯ g . Moreover, a path between b 0 ∈ ℳ g and b ∈ ∂ ⁡ ℳ g induces a map C b 0 → C b that shrinks disjoint loops of C b 0 to nodes of C b : call them shrinking loops.

Notation A.4.

If γ ⊂ C b 0 is a non-trivial simple loop, we denote by Tw γ ∈ MCG ⁢ ( C b 0 ) the right Dehn twist along γ (see [16, Chapter 3]). If γ 1 , … , γ k are non-trivial disjoint simple loops on C b 0 and a 1 , … , a k > 0 are integers, then Tw γ 1 a 1 ⁢ ⋯ ⁢ Tw γ k a k is the element of MCG ⁢ ( C b 0 ) obtained by performing a i Dehn twists along γ i for i = 1 , … , k .

The following statement is rather standard.

Lemma A.5 (Boundary strata and Dehn twists).

Let Y ¯ be an irreducible (not necessarily closed) analytic subvariety of M ¯ g , call Y := Y ¯ ∩ M g and let b ∈ ∂ ⁡ Y . Suppose b belongs to a locally-closed boundary stratum δ of M ¯ g . Fix a point b 0 ∈ Y sm and a path in Y sm ∪ { b } from b 0 to b. Let γ 1 , … , γ k ⊂ C b 0 be the induced shrinking loops. Then there exists a 1 , … , a k ≥ 1 such that the image of π 1 ⁢ ( Y sm , b 0 ) → MCG ⁢ ( C b 0 ) contains Tw γ 1 a 1 ⁢ ⋯ ⁢ Tw γ k a k . Hence, the image of π 1 ⁢ ( Y sm , b 0 ) → Sp ⁡ ( H 1 ⁢ ( C b 0 ) ) contains the transvection associated to the vector [ a 1 ⁢ γ 1 + … + a k ⁢ γ k ] .

Proof.

Pick a holomorphic map f : Δ → Y ¯ such that f ⁢ ( 0 ) = b and f ⁢ ( Δ * ) is contained in Y sm . Since Y sm is connected, we can choose b 0 as b 0 = f ⁢ ( b ~ 0 ) for some 0 ≠ b ~ 0 ∈ Δ . Every branch of ∂ ⁡ ℳ g that contains δ corresponds to a loop γ i on C b 0 . If f has multiplicity a i ≥ 1 along the i-th branch at b ~ 0 , the image of π 1 ⁢ ( Δ * , b ~ 0 ) → MCG ⁢ ( C b 0 ) is generated by ∏ i = 1 k Tw γ i a i , and the conclusion follows. ∎

The claimed result is an easy consequence of the above lemma.

Proof of Proposition A.3.

Let b 2 be a point in the boundary of Y ¯ ∩ δ h irr . Pick a contractible neighborhood U of b 2 inside ℳ ¯ g such that U ∩ δ ¯ h irr and U ∩ Y ¯ are contractible too. Let b 0 ∈ U ∩ Y sm and b 1 ∈ U ∩ δ h irr . Connect b 0 to b 1 and to b 2 through paths contained in Y sm (except at b 1 , b 2 ). Call β 1 , … , β k the disjoint loops in C b 0 that are shrunk by the map C b 0 → C b 1 . Up to isotopy, we can assume that the map C b 0 → C b 2 shrinks β 1 , … , β k and the other loops β k + 1 , … , β m to nodes. By Lemma A.5 the image of π 1 ⁢ ( Y sm ) → Γ g contains the commuting transvections T v , T w , where v = a 1 ⁢ β 1 + ⋯ + a k ⁢ β k and w = c 1 ⁢ β 1 + … + c m ⁢ β m with a i , c i ≥ 1 . Since h ≥ 1 , at least one loop β 1 , … , β k is non-disconnecting and so the vector v is non-zero. Since b 2 ∈ δ ¯ h + 1 irr , there exists a loop β i with k + 1 ≤ i ≤ m such that C b 0 ∖ ( β 1 ∪ … ∪ β k ) and C b 0 ∖ ( β 1 ∪ … ∪ β k ∪ β i ) have the same number of connected components. It follows that w is not a multiple of v, and so v , w are linearly independent. ∎

We now want to use Proposition A.3 to obtain a similar statement but for subvarieties of 𝒜 g ⁢ ( Γ ) instead of ℳ g , where Γ is any finite-index subgroup of Γ g .

We begin with an elementary lemma.

Lemma A.6.

Let ι : Z → A g the inclusion of a subvariety and let N , M be subgroups of π 1 ⁢ ( Z ) such that [ N : M ∩ N ] < + ∞ . Denote by ι * : π 1 ⁢ ( Z ) → π 1 ⁢ ( A g ) = Γ g the homomorphism induced by ι. If ι * ⁢ ( N ) contains a pair of linearly independent, commuting transvections, then ι * ⁢ ( M ) does.

Proof.

Let α , β ∈ N such that T v = ι * ⁢ ( α ) , T w = ι * ⁢ ( β ) ∈ ι * ⁢ ( N ) are two linearly independent, commuting transvections. There exists an integer m ≥ 1 such that α m , β m ∈ M ∩ N . It follows that T m ⁢ v = ι * ⁢ ( α m ) , T m ⁢ w = ι * ⁢ ( β m ) is the wished pair of linearly independent, commuting transvections in ι * ⁢ ( M ) . ∎

In the below lemma we prove the wished statement for subvarieties of 𝒜 g .

Lemma A.7 (Finding a pair of commuting transvections II).

Let Z ⊂ A g be an irreducible subvariety that meets the Jacobian locus, and suppose that Z ¯ ∩ J ¯ g meets the locally-closed boundary stratum A g - h in a non-compact subset. Then the image of π 1 ⁢ ( Z sm ) → Γ g contains two linearly independent, commuting transvections.

Proof.

Since Z ¯ ∩ 𝒥 ¯ g inside 𝒜 ¯ g meets the boundary stratum 𝒜 g - h in a non-compact subset, the closure of t - 1 ⁢ ( Z ) inside ℳ ¯ g meets the locally-closed boundary stratum δ h irr in a non-compact subset. Thus, an irreducible component Y of t - 1 ⁢ ( Z ) satisfies hypotheses (a) and (b) in Proposition A.3, and so the image of π 1 ⁢ ( Y sm ) → Γ g contains two linearly independent, commuting transvections.

Call H the image of π 1 ⁢ ( Y sm ) → π 1 ⁢ ( Z ) and K the image of π 1 ⁢ ( Z sm ) → π 1 ⁢ ( Z ) . Since Y → Z is a finite map and Y , Z are irreducible, Corollary 2.7 (i) implies [ H : K ∩ H ] < + ∞ . By Lemma A.6 applied to N = H and M = K , the image of π 1 ⁢ ( Z sm ) → Γ g contains two linearly independent, commuting transvections. ∎

As a consequence, we obtain our criterion for subvarieties of 𝒜 g ⁢ ( Γ ) .

Corollary A.8 (Finding a pair of commuting transvections III).

Let Γ be a finite-index subgroup of Γ g , let p : A g ⁢ ( Γ ) → A g be the natural projection and let Z ⊂ A g ⁢ ( Γ ) be an irreducible subvariety. Suppose that the image p ⁢ ( Z ) meets the Jacobian locus inside A g , and that p ⁢ ( Z ) ¯ ∩ J ¯ g intersects the boundary stratum A g - h in a non-compact subset. Then the image of π 1 ⁢ ( p ⁢ ( Z ) sm ) → Γ g contains two linearly independent, commuting transvections.

Proof.

Observe that p ⁢ ( Z ) is irreducible, and let ι : p ⁢ ( Z ) → 𝒜 g be the inclusion and ι * : π 1 ⁢ ( p ⁢ ( Z ) ) → Γ g the induced homomorphism. Moreover, let us denote by H the image of π 1 ⁢ ( Z sm ) → π 1 ⁢ ( p ⁢ ( Z ) ) and by K the image of π 1 ⁢ ( p ⁢ ( Z ) sm ) → π 1 ⁢ ( p ⁢ ( Z ) ) .

By Lemma A.7, the subgroup ι * ⁢ ( K ) contains two linearly independent, commuting transvections. Since Z → p ⁢ ( Z ) has finite fibers, it follows that H is a finite-index subgroup of K by Corollary 2.7 (ii) in the case Y = Z . Hence, ι * ⁢ ( H ) is a finite-index subgroup of ι * ⁢ ( K ) , and so it contains two linearly independent, commuting transvections by Lemma A.6 applied to N = K and M = H . ∎

Acknowledgements

We are indebted to Eberhard Freitag for stimulating discussions, suggestions and many useful comments on an earlier version of the manuscript. We thank Mark Goresky for useful discussions on Lefschetz hyperplane section theorem and Julia Bernatska for illustrating her result on gradients of theta functions. We are also grateful to an anonymous referee for carefully reading the paper and for valuable suggestions.

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Received: 2021-12-12

Revised: 2023-07-04

Published Online: 2023-11-20

Published in Print: 2024-01-01

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