Differential Algebraic Framework for Complex Geometry: Constructive Methods for Holomorphic Structures and Geometric PDEs

Differential Algebraic Framework for Complex Geometry: Constructive Methods for Holomorphic Structures and Geometric PDEs | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 28 October 2025 V1 Latest version Share on Differential Algebraic Framework for Complex Geometry: Constructive Methods for Holomorphic Structures and Geometric PDEs Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176168568.82374640/v1 184 views 109 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper develops a comprehensive differential algebraic frame work for complex geometry, extending the methods of differential algebraic closures to holomorphic structures, K¨ahler geometry, and complex geometric PDEs. We construct the complex geometric closure KComplex through a recursive adjunction process that incorporates holomorphic functions, Dolbeault cohomology classes, K¨ahler metrics, and solutions to complex Monge-Amp`ere equations. Within this closure, we prove explicit representation theorems for complex structures, holomorphic vector bundles, and geometric objects on complex manifolds. The framework provides constructive solutions to fundamental problems in complex geometry, including the existence of K¨ahler-Einstein metrics, classification of holomorphic vector bundles, and deformation theory of complex structures. We establish rigorous convergence criteria in appropriate complex-analytic function spaces and provide detailed algorithms with certified error bounds using complex interval arithmetic. The work demonstrates that explicit analytic solutions to complex geometric problems exist within appropriately constructed differential algebraic closures, offering new algebraic perspectives on classical complex geometry while maintaining consistency with established theory. Supplementary Material File (complex_geometry.pdf) Download 356.71 KB Information & Authors Information Version history V1 Version 1 28 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords complex geometry complex monge-ampère equation constructive mathematics deformation theory differential algebra dolbeault cohomology holomorphic vector bundles kähler manifolds Authors Affiliations Dongqi Liu 0009-0006-4018-9292 View all articles by this author shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 184 views 109 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Differential Algebraic Framework for Complex Geometry: Constructive Methods for Holomorphic Structures and Geometric PDEs. Authorea . 28 October 2025. 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