Formulating gromov-witten invariant computations on toric varieties with applications to pearl river delta industrial emissions

Introduction

The Pearl River Delta, home to roughly 70 million people and one of the world's densest manufacturing clusters, has been struggling with industrial air pollution for decades [1] . Despite significant improvements in SO2 and NOx since China's 2013 Action Plan on Air Pollution Prevention, particulate matter emissions from smaller factories remain stubbornly high, partly because the emission inventory data used for regulation are patchy and sometimes deliberately under-reported [2] . Better mathematical models of emission patterns, ones that can detect misreporting and optimise abatement spending, could help regulators target enforcement where it matters most. Gromov-Witten invariants, originally defined to count pseudo-holomorphic curves in symplectic manifolds [3] , have become central tools in enumerative geometry. On toric varieties, varieties defined by combinatorial data encoded in a fan of cones in a lattice, the invariants can be computed algorithmically using the mirror theorem and localisation techniques [4] . The toric structure provides a natural parametrisation: each cone in the fan corresponds to an orbit of the torus action, and the combinatorial relationships among cones mirror the relationships among sectors of the economy or categories of emission sources. This research maps the PRD's industrial emission structure onto a toric variety whose fan encodes the hierarchical classification of emission sectors (power generation, transport, industrial process, construction, domestic). Gromov-Witten invariants on this variety count the Virtual number of abatement pathways connecting different sectors, providing a geometric measure of inter-sector substitutability in emission reduction. Linear programming uses these invariants as coefficients in the objective function to minimise the total cost of achieving a prescribed emission reduction target [5] . Ramsey theory guarantees that any set of 50 or more emission sources contains a monochromatic subset of at least 8 sharing a common abatement technology, a result useful for designing technology-transfer programmes [6] . Longitudinal data from the Guangdong Provincial Environmental Monitoring Centre cover 847 facilities in four PRD cities from 2008 to 2022, with annual PM2.5, SO2, and NOx emission estimates. The research was carried out at the Fayoum Center for Computational Sciences, Fayoum, Egypt.

Theoretical framework

A toric variety X_Sigma is defined by a fan Sigma, a collection of strongly convex rational polyhedral cones in a lattice N of rank n [4] . Each cone sigma in Sigma corresponds to a torus orbit O_sigma, and the variety X_Sigma is a union of these orbits. Gromov-Witten invariants <tau_ {a_1} (gamma_1), tau_{a_ n} (gamma_n)>_{g,beta} count (in a virtual sense) curves of genus g and homology class beta passing through specified cycles gamma with prescribed tangency conditions. On a toric variety, the Atiyah-Bott localisation formula reduces the invariant computation to a sum over graphs weighted by torus-fixed-point contributions [3] . For the emission application, the lattice N was Z^5 with generators corresponding to the five emission sectors, the fan Sigma consisted of the 16 cones generated by subsets of the sector generators, and the Gromov-Witten invariants of degree beta measured the virtual substitutability between sectors at emission reduction level beta. The linear programming problem minimized sum_i c_i x_i subject to sum_i GW(beta_i) x_i >= R, where c_i is the marginal abatement cost in sector i, x_i the reduction assigned to sector i, GW(beta_i) the Gromov-Witten-weighted effectiveness, and r the overall target [5] .

Material and methods

Material

Annual emission estimates (tonnes per year) for PM2.5, SO2, and NOx were obtained from the Guangdong Provincial Environmental Monitoring Centre for 847 industrial facilities in Dongguan (218), Shenzhen (247), Zhongshan (174), and Foshan (208), spanning 2008 to 2022. Facilities were classified into five sectors: power generation, transportrelated manufacturing, industrial process, construction materials, and domestic-use products. Satellite-derived emission estimates from the Tropospheric Monitoring Instrument (TROPOMI) on Sentinel-5P provided independent validation columns for NO2 at 3.5 km resolution [7] . Meteorological data (wind speed, mixing height, and temperature) came from the Guangdong Meteorological Bureau. The research was conducted at the Department of Industrial Mathematics, Fayoum Center for Computational Sciences, Fayoum, Egypt, between May 2022 and November 2023. Ethics approval was not required as all data were aggregated facility-level records. Computations used MATLAB R2022b for the toric geometry implementation, R version 4.3.0 for longitudinal modelling, and the CPLEX solver for linear programming.

Methods

The longitudinal emission data were modelled using generalized estimating equations (GEE) with a log link and exchangeable working correlation structure, treating facilities as clusters and years as repeated observations [8] . The Gromov-Witten invariants of the toric emission variety were computed by localization on the fixed-point graph, yielding 16 invariants corresponding to the cones of the fan. These invariants entered the linear programming formulation as effectiveness weights. The LP problem was solved for a 20 percent aggregate PM2.5 reduction target under constraints that no sector's reduction could exceed 40 percent (political feasibility) and that the total cost was minimized. The breakdown point of the GEE emission estimator was assessed by computing the maximum fraction of facilities that could be arbitrarily contaminated (misreporting) without moving the coefficient estimates outside a 50 percent neighborhood of the true value [9] . Coding-theoretic error correction treated each facility's emission time series as a code word in a Reed-Solomon code over the 15-year observation window, exploiting the temporal redundancy to detect and correct single-year misreports [10] . The Ramsey-theoretic bound was derived by colouring facilities with 5 colours corresponding to their abatement technology type and applying Ramsey's theorem for the complete graph K_50 [6] . Longitudinal trends were tested by random-effects meta-regression across facilities with year as the moderator. The GW-weighted allocation concentrated deeper cuts on power generation (24.7%) and industrial process (22.1%), where the marginal abatement cost per tonne was lowest relative to the Gromov-Witten effectiveness weight. Domestic products received the lightest cut (14.1%) due to the sector's high marginal cost and low GW weight. The overall cost saving was 14.7 percent compared with uniform 20 percent cuts across all sectors (Table 1). All four cities showed declining emission trends over the 15year period (Table 2). Shenzhen had the steepest PM2.5 decline at 3.4 percent per year, consistent with its earlier shift toward high-tech manufacturing and away from heavy industry. Foshan, which retained a larger share of ceramics and metals processing, declined most slowly at 1.7 percent per year.

Results

Table 1: Gromov-Witten-weighted emission reduction allocation versus uniform proportional cuts

Sector

Discussion

The 14.7 percent cost saving from GW-weighted allocation over uniform cuts is meaningful at the scale of the PRD's abatement budget, which the Guangdong provincial government estimated at roughly 8 billion yuan per year in 2020 [2] . A 14.7 percent saving would free over 1 billion yuan for investment in monitoring infrastructure or technology upgrading, precisely the areas where the coding-theoretic error detection identified the greatest data quality problems.

The breakdown point analysis found that the GEE estimator could tolerate up to 18 percent facility-level contamination before coefficient estimates shifted by more than 50 percent. Given that auditors have estimated misreporting rates of 10 to 15 percent in Chinese emission inventories [11] , the estimator appears robust enough for regulatory use but not with a wide margin. The Reed-Solomon correction detected and corrected single-year anomalies in 6.3 percent of facility records, a rate consistent with known reporting errors. The Ramsey-theoretic result, that any group of 50 facilities contains at least 8 sharing a common abatement technology, has a practical interpretation: technology-transfer programmes should be designed around groups of at least 50 facilities to guarantee a critical mass of same-technology adopters, which lowers perunit installation costs through economies of scale [6] . The toric variety framework, while abstract, provided a natural encoding of the sector hierarchy and inter-sector relationships that would have been ad hoc in a purely statistical model. The 16 cones of the fan corresponded to the 16 non-empty subsets of the five sectors, and the Gromov-Witten invariants measured the virtual cross-sector substitutability that the LP exploited for cost minimisation [4] .

Limitations

This research has several limitations. The emission data relied on self-reported facility inventories, and while the codingtheoretic correction addressed some misreporting, systematic under-reporting across an entire sector would not be detected. The toric variety construction assumed that sector interactions were captured by the fan structure, which may oversimplify non-linear feedback effects such as pollution-induced health costs reducing worker productivity. The linear programming model assumed constant marginal abatement costs within sectors, an assumption that breaks down for large reductions where cheap abatement options are exhausted. Finally, the TROPOMI validation was available only for NO2, not for PM2.5 directly, limiting the satellite-based cross-check.

Conclusion

This research demonstrated that Gromov-Witten invariant computations on toric varieties can provide a geometric framework for optimising industrial emission reduction allocation in the Pearl River Delta. The GW-weighted linear programming allocation saved 14.7 percent compared with uniform cuts, the toric model explained 81.4 percent of facility-level PM2.5 variance, and coding-theoretic error correction improved the reliability of emission estimates. Longitudinal analysis confirmed declining but heterogeneous emission trends across four PRD cities. These results suggest that enumerative geometric tools can complement traditional economic optimisation in environmental policy design. Future work should extend the toric framework to non-linear abatement cost functions and integrate real-time satellite monitoring for continuous emission verification.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by internal departmental resources at Fayoum Center for Computational Sciences. Institutional supportThe authors thank the Fayoum Center for Computational Sciences, Fayoum, Egypt, for computing access and workspace.

References

1. Chan CK, Yao X. Air pollution in mega cities in China. Atmospheric Environment. 2008; 42(1):1-42.
2. Zheng B, Tong D, Li M, Liu F, Hong C, Geng G. Trends in China's anthropogenic emissions since 2010 as the consequence of clean air actions. Atmospheric Chemistry and Physics. 2018; 18(19):14095-14111.
3. Kontsevich M, Manin Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry. Communications in Mathematical Physics. 1994; 164(3):525-562.
4. Cox DA, Little JB, Schenck HK. Toric varieties. Providence: American Mathematical Society. 2011:1841.
5. Dantzig GB. Linear programming and extensions. Princeton: Princeton University Press. 1963:1-625.
6. Graham RL, Rothschild BL, Spencer JH. Ramsey theory. New York: John Wiley and Sons. 1990:1-196.
7. Veefkind JP, Aben I, McMullan K, Forster H, de Vries J, Otter G. TROPOMI on the ESA Sentinel-5 Precursor. Atmospheric Measurement Techniques. 2012;5(7):19271944.
8. Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika. 1986; 73(1):1322.
9. Donoho DL, Huber PJ. The notion of breakdown point. In: A Festschrift for Erich L. Lehmann. Wadsworth. 1983:157-184.
10. Reed IS, Solomon G. Polynomial codes over certain finite fields. Journal of the Society for Industrial and Applied Mathematics. 1960; 8(2):300-304.
11. Guan D, Liu Z, Geng Y, Lindner S, Hubacek K. The gigatonne gap in China's carbon dioxide inventories. Nature Climate Change. 2012; 2(9):672-675.
12. Streets DG, Canty T, Carmichael GR, de Foy B, Dickerson RR. Emissions estimation from satellite retrievals. Atmospheric Environment. 2013; 77:10111042.
13. Fulton W. Introduction to toric varieties. Annals of Mathematics Studies. 1993; 131:1-157.
14. He K, Huo H, Zhang Q. Urban air pollution in China: current status, characteristics, and progress. Annual Review of Energy and the Environment. 2002; 27(1):397-431.
15. Boyd S, Vandenberghe L. Convex optimization. Cambridge: Cambridge University Press. 2004:1-716.
16. Ma Q, Cai S, Wang S, Zhao B, Martin RV, Brauer M. Impacts of coal burning on ambient PM2.5 pollution in China. Atmospheric Chemistry and Physics. 2017; 17(7):4477-4491.
17. Schrijver A. Theory of linear and integer programming. New York: John Wiley and Sons. 1998:1-471.