Positivity of Landau–Ginzburg TERP Structures

Definitions and mathematical statement

Let $Y$ be a smooth affine algebraic manifold over ${\mathbb{C}}$ and let $f:Y\to{\mathbb{C}}$ be algebraic, with only isolated singularities and tame at infinity. The source asks whether the TERP structure associated to $f$ by the Landau--Ginzburg construction is positive-definite, pure and polarized. The tameness condition and the exact construction conventions are referenced but not defined in the supplied record.

A TERP structure, with its specified weight $w$, consists schematically of a holomorphic bundle $H$ over the $z$-plane, a flat connection on $H|_{{\mathbb{C}}^*}$ with the permitted pole at $z=0$, a flat real form there, and a nondegenerate flat pairing relating fibers at $z$ and $-z$, with prescribed weight, symmetry, real and extension conditions. In the usual setup the connection has pole order at most two at zero. The Landau--Ginzburg object uses the relevant Gauss--Manin/Fourier--Laplace and Brieskorn-lattice data, not an arbitrary bundle chosen to satisfy positivity. Its precise weight, real structure and pairing normalization must be retained from that construction.

To explain purity and positivity, the real structure and connection determine a twistor gluing to a bundle $\widehat H$ over ${\mathbb{P}}^1$. In the usual weight-normalized TERP convention, purity means \[ \widehat H\simeq\mathcal O_{{\mathbb{P}}^1}^{\oplus r},\qquad r=\operatorname{rank}H. \] The normalized pairing induces a Hermitian form $h$ on the corresponding space of global sections. Polarization/positive definiteness requires \[ h(s,s)>0\quad\text{for every nonzero }s\in H^0({\mathbb{P}}^1,\widehat H). \] The sign and weight factors defining $h$ cannot be chosen independently of the source construction. This describes the intended properties while leaving missing normalizations explicit.

The conjecture concerns the TERP structure of the original, unscaled function and its specified data. It is not replaced by a statement holding only after rescaling $f$ or the connection parameter, or only in an asymptotic regime. The supplied literature assessment checked 21 August 2026 reports scaled, asymptotic and special-case results in later work, but says its reviewer located no proof of the exact unscaled claim. That is a dated supplied assessment, not an independent current-status conclusion.

Short English statement

For a tame affine Landau--Ginzburg function with isolated singularities, is its original TERP structure pure and positively polarized? The relevant twistor bundle should be trivial and its induced Hermitian form positive. Scaled or asymptotic positivity, reported as partial progress, is not the same assertion.

Sources

• Local UnsolvedMath registry, permanent ID 30000105, exact source code catalogue entry; Positivity of Landau--Ginzburg TERP Structures. Proposed year 2004, status open; corrected_verified statement makes the affine, isolated-singularity and tameness hypotheses explicit.
• Supplied report citation: String-Theorie und Geometrie (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/38, pp. 2015--2070, https://doi.org/10.4171/owr/2004/38. The title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21: "OWR report", https://doi.org/10.4171/owr/2004/38; "Hertling--Sevenheck", https://arxiv.org/abs/math/0603564. This dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.