[Paper Review] Gevrey solutions of irregular hypergeometric systems in two variables

María‐Cruz Fernández‐Fernández, Francisco Jesús Castro Jiménez|arXiv (Cornell University)|Nov 20, 2008

Polynomial and algebraic computation12 references4 citations

TL;DR

This paper provides a complete description of Gevrey series solutions at singular points for irregular hypergeometric systems in two variables associated with affine plane monomial curves. Using Γ-series and elementary D-module techniques, it proves that the irregularity complex of such systems is a perverse sheaf and identifies the analytic slope as $ b/a $, confirming the slope comparison theorem in a specific case.

ABSTRACT

We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine plane monomial curve. We also describe the irregularity complex of such a system with respect to its singular support.

Motivation & Objective

• To explicitly describe Gevrey series solutions of irregular hypergeometric systems $ \mathcal{M}_A(\beta) $ at singular points for $ A = (a_1\; a_2) $ with $ \gcd(a_1,a_2) = 1 $.
• To analyze the irregularity complex $ \operatorname{Irr}_Y(\mathcal{M}_A(\beta)) $ with respect to the singular support $ Y $, the union of coordinate axes.
• To prove, using elementary methods, that the irregularity complex is a perverse sheaf on $ Y $, confirming a general result of Mebkhout.
• To identify the analytic slope of the system with respect to the coordinate axes as $ b/a $, and verify it matches the algebraic slope.
• To provide a basis for the 0-th cohomology of the irregularity complex at non-origin points on $ Y $, and describe monodromy eigenvalues explicitly.

Proposed method

• Utilizes Γ-series, introduced by Gel’fand, Kapranov, and Zelevinsky, to parametrize holomorphic solutions at generic points.
• Applies the formalism of Gevrey series of order $ s $ along a smooth curve $ Y $, defined via convergence of $ \sum \frac{1}{i!^{s-1}} f_i(x_1) x_2^i $.
• Constructs the irregularity complex $ \operatorname{Irr}_Y^{(s)}(\mathcal{M}_A(\beta)) $ as the hypercohomology of a complex involving $ \mathcal{O}_{\widehat{X|Y}}(s) $ and $ \mathcal{Q}_Y(s) $.
• Employs the long exact sequence in $ \mathcal{E}xt $-cohomology for $ \mathcal{D}_X $-modules to analyze the structure of $ \operatorname{Irr}_Y^{(s)}(\mathcal{M}_A(\beta)) $ at points $ p \in Y $.
• Uses the operator $ E = a x_1 \partial_{x_1} + b x_2 \partial_{x_2} - \beta $ and its action on formal series to derive the structure of solutions.
• Applies the monodromy computation via eigenvalues $ \exp\left(\frac{2\pi i(\beta - b k)}{a}\right) $ for $ k = 0, \dots, a-1 $ to verify the co-support condition for perversity.

Experimental results

Research questions

• RQ1 What is the structure of Gevrey series solutions of the irregular hypergeometric system $ \mathcal{M}_A(\beta) $ at the origin for $ A = (a_1\; a_2) $?
• RQ2 How does the irregularity complex $ \operatorname{Irr}_Y(\mathcal{M}_A(\beta)) $ behave at the origin and at non-singular points of the singular support $ Y $?
• RQ3 Is the irregularity complex of $ \mathcal{M}_A(\beta) $ a perverse sheaf on $ Y $, and can this be shown without deep D-module theory?
• RQ4 What is the analytic slope of $ \mathcal{M}_A(\beta) $ with respect to the coordinate axes, and how does it compare to the algebraic slope?
• RQ5 Can a basis for the 0-th cohomology of the irregularity complex be explicitly described at points $ p \in Y \setminus \{(0,0)\} $?

Key findings

• The germ of the irregularity complex $ \operatorname{Irr}_Y(\mathcal{M}_A(\beta)) $ at the origin is zero, as shown in Theorem 4.1.
• For $ p \in Y \setminus \{(0,0)\
• The irregularity complex $ \operatorname{Irr}_Y^{(s)}(\mathcal{M}_A(\beta)) $ is concentrated in degree 0 for $ b/a \leq s \leq \infty $, and zero for $ 1 \leq s < b/a $, establishing a unique gap at $ s = b/a $.
• The 0-th cohomology of the irregularity complex at $ p \in Y \setminus \{(0,0)\
• The monodromy eigenvalues of the irregularity complex are $ \exp\left(\frac{2\pi i(\beta - b k)}{a}\right) $ for $ k = 0, \dots, a-1 $, which are explicitly computed from the basis of solutions.
• The analytic slope of $ \mathcal{M}_A(\beta) $ with respect to $ Y $ is $ b/a $, and this matches the algebraic slope, confirming the slope comparison theorem in this case.