[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"mU9203UUea":3},"# Several Complex Variables\n\nA Lean 4 formalization of the classical theory of several complex variables, built on\nMathlib, with pinned TauCeti modules available as an additional source of results, and\nintended as a contribution to Mathlib. The subject is function theory on open subsets\nof finite-dimensional complex normed spaces, in particular domains in ℂⁿ: Cauchy and Taylor\ntheory, convergence and function spaces, local holomorphic mappings, Reinhardt and Laurent\ncontinuation, the Hartogs phenomena and removable singularities, analytic sets, analytic germs\nwith Weierstrass theory, holomorphic convexity with the Cartan–Thullen theorem and Bochner's\ntube theorem, plurisubharmonic functions and Levi convexity, and the elementary theory of\nRunge domains. Statements are Banach-valued wherever the arguments allow.\n\nThe mathematical content is described in **[SCVMainTheorems.md](SCVMainTheorems.md)**: the\ndefinitions and conventions, the catalogue of 65 principal results with proof sketches, and\nthe extent and limits of the theory. Every catalogued result is proved.\n\nThe first version of the project has been entered into the\n[Palomar Registry](https://palomar-registry.org/) as\n[PALOMAR-2026-09-18-000008 v1](https://palomar-registry.org/entry?id=PALOMAR-2026-09-18-000008&version=1).\nThat entry has 63 theorems. The current `Challenge.lean` restates 67 theorems with\nintentional `sorry` placeholders; their proofs are supplied in `Solution.lean`.\n\n## Organization\n\n- `SeveralComplexVariables.lean` is the root module. It imports every file of the library\n  and its docstring summarizes the contents topic by topic.\n- `SeveralComplexVariables/` holds the library, grouped by topic (`AnalyticGerm`,\n  `AnalyticSet`, `HolomorphicConvexity`,\n  `LaurentSeries`, `LeviConvexity`, `Reinhardt`, `RemovableSingularity`, `Runge`,\n  `SeparateAnalytic`, `TubeDomain`, `WeierstrassDivision`, `ZeroSets`, and others). Every\n  file has a documentation header and every declaration a docstring.\n- `ComplexAnalysis/` holds the single-variable complex analysis used by the library (Cauchy\n  transforms and the Cauchy–Pompeiu identity, annulus Laurent series, subharmonic functions,\n  one-variable removable singularities, and related results). It is the subset of a separate\n  one-variable project that this library needs.\n- `ToMathlib/Algebra/`, `ToMathlib/Analysis/` and `ToMathlib/Topology/` hold general algebra,\n  analysis and topology support, in the namespaces of the Mathlib APIs they extend.\n  All three layers may import suitable Mathlib and TauCeti modules.\n- `Main.lean` is the executable stub required by the Lake configuration; it only imports the\n  library.\n- `Challenge.lean` states the selected Palomar results using Mathlib alone; `Solution.lean`\n  proves the matching statements using the library.\n- `formalization.yaml` records submission metadata and scope; `comparator.json` lists the\n  declarations to compare. `make_challenge.py` generates the challenge and Comparator\n  configuration from `Solution.lean`.\n- `SeveralComplexVariables/References/` (not in the repository) contains the texts that\n  guided the selection of material; see below.\n\n## Building the current project\n\nThe current working tree uses Lean `v4.36.0-rc1`, Mathlib revision\n`911003dd6dea5df215dff57f2717be762e11a6ca`, and TauCeti revision\n`5cdecf40e99d5a5ad0833afff31f5b007d68a149` (see `lean-toolchain` and `lakefile.toml`).\nThe published Palomar V1 snapshot used `v4.35.0-rc2`.\n\n\nThe 18 October 2026 upgrade uses Mathlib's development head. TauCeti's own\nMathlib pin is 65 commits earlier; [TauCeti-4.36.patch](TauCeti-4.36.patch)\nupdates one renamed coercion lemma for this snapshot. The shared checkout\nalready has this patch. After a fresh dependency checkout, apply it before\nbuilding:\n\n```sh\ngit -C .lake/packages/TauCeti apply \"$PWD/TauCeti-4.36.patch\"\n```\n\nPrefer existing results in Mathlib, then the pinned TauCeti, before developing local proofs.\nImport individual modules, for example\n`public import TauCeti.Analysis.Complex.Conformal.Inverse.Function`;\n`import TauCeti` does not re-export the library. Resolve TauCeti prerequisites when preparing\nMathlib contributions. See [TauCetiReview.md](TauCetiReview.md) for the reuse inventory.\n\nFrom the repository root:\n\n    lake build\n\nTo check both Palomar modules:\n\n    lake build Solution Challenge\n\nThe Challenge build reports the 67 intentional `sorry` placeholders. These build checks\nare separate from Palomar's Comparator and independent kernel verification.\n\nThe Palomar version is a snapshot. The Github version of the project may see further\ndevelopment.\n\n## Documentation\n\n- [SCVMainTheorems.md](SCVMainTheorems.md): definitions, conventions, and the catalogue of\n  principal results with their proof status. This is the mathematical description of the\n  project.\n\n## Conventions in brief\n\nHolomorphic means `DifferentiableOn ℂ` on an open set and analytic means `AnalyticOnNhd ℂ`;\ntheir equivalence is proved in the setting used. The norm on `ι → ℂ` is the supremum norm,\nso its balls are polydiscs; Euclidean balls are identified explicitly. Hulls are defined by\nall real upper bounds, subharmonicity by the local submean property, and the Cauchy–Riemann\noperator by the antiholomorphic part of a real derivative, without differential forms. The\ndeferred targets are the Oka–Weil approximation theorem, the one-variable Runge theorem for\nrational approximation, and the Levi problem in the sufficiency direction.\n\n## References\n\nThe module docstrings use Mathlib-style bibliography keys resolved in\n[SCVReferences.bib](SCVReferences.bib). These entries accompany the project modules for\nupstream integration; the installed Mathlib bibliography is unchanged.\n\nThe formalization follows classical introductory treatments, especially:\n\n- H. Alexander and J. Wermer, *Several Complex Variables and Banach Algebras*, 3rd ed.,\n  Springer GTM 135, 1998.\n- H. P. Boas, *Lecture Notes on Several Complex Variables*, manuscript, 2013.\n- K. Fritzsche and H. Grauert, *From Holomorphic Functions to Complex Manifolds*, Springer\n  GTM 213, 2002.\n- L. Hörmander, *An Introduction to Complex Analysis in Several Variables*, 2nd ed.,\n  North-Holland, 1973.\n- P. Jakóbczak and M. Jarnicki, *Lectures on Holomorphic Functions of Several Complex\n  Variables*, manuscript, 2021.\n- J. Korevaar and J. Wiegerinck, *Several Complex Variables*, version of 23 August 2017.\n- S. G. Krantz, *Function Theory of Several Complex Variables*, 2nd ed., 1992.\n- J. Lebl, *Tasty Bits of Several Complex Variables*, 2026.\n- P. Lelong and L. Gruman, *Entire Functions of Several Complex Variables*, Springer, 1986.\n- R. M. Range, *Holomorphic Functions and Integral Representations in Several Complex\n  Variables*, Springer GTM 108, 1986.\n- V. Scheidemann, *Introduction to Complex Analysis in Several Variables*, Birkhäuser, 2005.\n- B. V. Shabat, *Introduction to Complex Analysis, Part II: Functions of Several Variables*,\n  1991.\n- T. Suwa, *Complex Analytic Geometry: From the Localization Viewpoint*, World Scientific,\n  2024.\n\n## License\n\nApache License 2.0; see [LICENSE](LICENSE).\n",1793473257034]