[Paper Review] Automorphic vector bundles with global sections on $G$-${ t Zip}^{\mathcal Z}$-schemes

Wushi Goldring, Jean-Stefan Koskivirta|arXiv (Cornell University)|Jan 2, 2017

Advanced Algebra and Geometry17 references3 citations

TL;DR

This paper proposes a conjecture on the cone of automorphic vector bundles with global sections on $G$-zip schemes of connected-Hodge-type, proving it for groups of type $A_1^n$, $C_2$, and $\mathbf{F}_p$-split $A_2$. It establishes that global sections are controlled by the geometry of the $G$-zip stack and the classifying morphism, with a counterexample showing the conjecture fails for non-connected-Hodge-type zip data.

ABSTRACT

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected-Hodge-type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_1^n$, $C_2$ and $\mathbf F_p$-split groups of type $A_2$ (this includes all Hilbert-Blumenthal varieties and should also apply to Siegel modular threefolds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected-Hodge-type.

Motivation & Objective

• To understand the extent to which the global geometry of schemes with a smooth surjective morphism to a $G$-zip stack $\GZip^\Zcal$ is controlled by the stack and the morphism.
• To formulate and prove a general conjecture on the cone of automorphic vector bundles admitting nonzero global sections on $G$-zip schemes of connected-Hodge-type.
• To establish the conjecture for specific groups: $A_1^n$, $C_2$, and $\mathbf{F}_p$-split $A_2$, including all Hilbert-Blumenthal varieties.
• To demonstrate via a counterexample that the conjecture fails when the zip data is not of connected-Hodge-type.
• To clarify the relationship between the cones $C_Y$, $C_\mathcal{Y}$, and $C_{\operatorname{Sbt}}$ in the context of automorphic line bundles and global sections.

Proposed method

• Leverages the theory of $G$-zips and their associated stacks $\GZip^\Zcal$ arising from Shimura varieties and Hodge theory.
• Applies the framework of automorphic vector bundles via the construction $\mathscr{V}(\lambda)$ associated to characters $\lambda$ in the weight lattice.
• Uses the cone $C_Y$ of characters with nonzero global sections and compares it to the cone $C_\mathcal{Y}$ derived from the $G$-zip structure.
• Employs the pullback of characters via the inclusion $\iota: \tilde{G} \to G$ to analyze the image of the weight lattice in the dual of the cocharacter group.
• Applies Kempf’s vanishing theorem and ampleness criteria to deduce the existence of global sections for certain line bundles.
• Uses matrix inversion and linear inequalities to describe the cone $C'_{\mathcal{Y}}$ in the dual space, proving non-membership of certain characters.

Experimental results

Research questions

• RQ1 To what extent is the global geometry of a scheme $X$ with a smooth surjective morphism $\zeta: X \to \GZip^\Zcal$ controlled by the stack $\GZip^\Zcal$ and the map $\zeta$?
• RQ2 What is the precise relationship between the cone $C_Y$ of characters with global sections and the cone $C_\mathcal{Y}$ derived from the $G$-zip structure?
• RQ3 Does the conjecture that $C_Y = C_\mathcal{Y}$ hold for all $G$-zip schemes of connected-Hodge-type?
• RQ4 Can the conjecture be shown to fail for zip data not of connected-Hodge-type, and if so, how?
• RQ5 How do the cones $C_{\operatorname{Sbt}}$, $C_\mathcal{Y}$, and $C_Y$ compare in the case of $G = GSp(6)$ and $X = \mathcal{A}_{3,K}$?

Key findings

• The conjecture that $C_Y = C_\mathcal{Y}$ holds for groups of type $A_1^n$, $C_2$, and $\mathbf{F}_p$-split $A_2$, including all Hilbert-Blumenthal varieties.
• For $G = GSp(6)$, the cone $C_Y$ strictly contains $C_{\operatorname{Sbt}}$, showing that the cone of global sections is not captured by sections pulled back from the special fiber.
• The counterexample with $G = GSp(6)$ and $X = \mathcal{A}_{3,K}$ shows that $C_Y \neq C_\mathcal{Y}$ when the zip data is not of connected-Hodge-type.
• The character $\lambda_n$ with $\iota^*(\lambda_n) = (-(p+1+n), p^2+n, p^2+1+n)$ has no global sections for any $n, m \geq 1$, so $\lambda_n \notin C_\mathcal{Y}$.
• Despite $\lambda_n \notin C_\mathcal{Y}$, the line bundle $\mathscr{L}_Y(\lambda_n)$ is ample for large $n$, implying $\lambda_n \in C_Y$, thus $C_Y \neq C_\mathcal{Y}$.
• The cone $C_\mathcal{Y}$ is strictly larger than $C_{\operatorname{Sbt}}$ in the $G = GSp(6)$ case, indicating a more complex structure than previously assumed.