[Paper Review] Spectra and strains

Vasily Golyshev|ArXiv.org|Jan 2, 2008

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper introduces the concepts of Fano spectra and strains as invariants for Fano varieties, linking quantum cohomology to arithmetic geometry via Galois representations. It proposes that spectra—derived from quantum multiplication matrices—encode deep arithmetic and geometric data, with low ramified sheaves and special Laurent polynomials offering a pathway to classify these spectra, especially in minimal and cellular Fano varieties.

ABSTRACT

This is a blend of two informal reports on the activities of the seminar on Galois representations and mirror symmetry given at the Conference on classification problems and mirror duality at the Steklov Institute, in March 2006, and at the Seminar on Algebra, Geometry and Physics at MPI, in November 2007. We assess where we are on the issue of the spectra of Fano varieties, and state problems. We introduce higher dimensional irreducible analogues of dessins, the low ramified sheaves, and hypothesize that Fano spectra relate to their geometric conductors. We give a recipe to a physicist.

Motivation & Objective

• To define and formalize the spectrum of a Fano variety using quantum multiplication matrices and their specialization to Laurent polynomials.
• To introduce the notion of 'strains' as deformation classes closed under hyperplane sections, and to define the spectrum of a strain as invariant under multiplicative shifts.
• To investigate the role of progenitor varieties—those with no unsections—as minimal representatives of their strains.
• To explore the connection between Fano spectra and low ramified sheaves (LRS), hypothesizing that spectral components arise from geometric monodromy data.
• To investigate whether spectra can serve as fine invariants distinguishing different Fano strains, especially in the minimal and cellular cases.

Proposed method

• Define the spectrum of a Fano variety as the zero locus of the determinant of the quantum multiplication matrix by the hyperplane class, specialized to a Laurent polynomial ring via curve degrees.
• Introduce the anticanonical spectrum via specialization using the anticanonical class, and define the complete anticanonical spectrum by adding components at infinity.
• Use Givental’s Quantum Weak Lefschetz theorem to show spectral stability under hyperplane sections, adjusting for index 1 cases via a Givental constant shift.
• Generalize dessins d'enfants to higher dimensions via 'low ramified sheaves'—local systems with minimal monodromy—defined over rational function fields.
• Relate the spectra of minimal Fano threefolds to modular objects such as elliptic points on modular curves $X_0(N)/W_N$, and to Picard-Fuchs equations.
• Construct a zoo of special Laurent polynomials by stratifying the space of polynomials with given support to single out those with degenerate critical values, linking them to low ramified sheaves.

Experimental results

Research questions

• RQ1 To what extent does the spectrum of a Fano variety determine its deformation class (strain), especially in the minimal or cellular case?
• RQ2 Can two non-isomorphic cellular strains have identical spectra, or is the spectrum a complete invariant for such classes?
• RQ3 Are all irreducible components of the completed anticanonical spectrum of a Fano variety also components of some low ramified sheaf (LRS) spectrum?
• RQ4 Is there a structural correspondence between the combinatorics of affine cell decompositions in cellular Fano varieties and the arithmetic conductors of associated LRS?
• RQ5 Can the field of definition of spectral components be shown to have small discriminant, suggesting optimality in their arithmetic complexity?

Key findings

• The spectrum of the Grassmannian $G(2,5)$ consists of the two roots of $t^2 - 11t - 1$, indicating it is not a complete intersection in projective space.
• The spectrum of the strain of complete intersections in projective space is a single point defined over $\mathbb{Q}$, and the strain is cellular, minimal, and Tate.
• The spectrum of the blowup of $\mathbb{P}^3$ along $\mathbb{P}^1$ consists of two irreducible components, each with Galois group $S_3$ and discriminants $-23$ and $-31$, the smallest discriminants for weight 1 cusp forms.
• The spectra of all Grassmannians are defined over $\mathbb{Q}^{ab}$, and irreducible components can have Galois group $S_n$ for $n \geq 5$, showing non-trivial arithmetic complexity.
• For minimal Fano threefolds, spectra are related to modular objects: they correspond to elliptic points and cusp points on $X_0(N)/W_N$, where $N = (-K_F)^3 / (2d^2)$.
• The paper conjectures that generic minimal Fano varieties have spectra that are LRS spectra, based on the Picard-Fuchs nature of their counting equations and low ramification of associated local systems.