Complex analytic variety

A cone is not a complex manifold, but it is a complex analytic variety.

In mathematics, particularly differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold that allows the presence of singularities. Complex analytic varieties are locally ringed spaces that are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.

Complex analytic varieties are analogous to algebraic varieties. Roughly speaking, a complex analytic variety is a zero locus of a set of a complex analytic function, while an algebraic variety is a zero locus of a set of a polynomial function.

Definition

Denote the constant sheaf on a topological space with value $ℂ$ by $\underset{\_}{ℂ}$. A $ℂ$-space is a locally ringed space $(X,{𝒪}_X)$, whose structure sheaf is an algebra over $\underset{\_}{ℂ}$.

Choose an open subset $U$ of some complex affine space ${ℂ}^n$, and fix finitely many holomorphic functions $f_1,\dots ,f_k$ in $U$. Let $X=V(f_1,\dots ,f_k)$ be the common vanishing locus of these holomorphic functions, that is, $X=\{x\mid f_1(x)=\cdots =f_k(x)=0\}$. Define a sheaf of rings on $X$ by letting ${𝒪}_X$ be the restriction to $X$ of ${𝒪}_U/(f_1,\dots ,f_k)$, where ${𝒪}_U$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $ℂ$-space $(X,{𝒪}_X)$ is a local model space.

A complex analytic variety is a locally ringed $ℂ$-space $(X,{𝒪}_X)$ that is locally isomorphic to a local model space.

Morphisms of complex analytic varieties are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps. A structure sheaf may have nilpotent elements; if the structure sheaf is reduced, then the complex analytic space is called reduced.

An associated complex analytic space (variety) $X_h$ is such that:

Let X be a scheme of finite type over $ℂ$, and cover X with open affine subsets $Y_i=\mathrm{Spec}{A}_i$ ($X=\cup Y_i$) (Spectrum of a ring). Then each $A_i$ is an algebra of finite type over $ℂ$, and $A_i\simeq ℂ[z_1,\dots ,z_n]/(f_1,\dots ,f_m)$, where $f_1,\dots ,f_m$ are polynomials in $z_1,\dots ,z_n$, which can be regarded as a holomorphic functions on $ℂ$. Therefore, their set of common zeros is the complex analytic subspace $(Y_i{)}_h\subseteq ℂ$. Here, the scheme X is obtained by glueing the data of the sets $Y_i$, and then the same data can be used for glueing the complex analytic spaces $(Y_i{)}_h$ into a complex analytic space $X_h$, so we call $X_h$ an associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space $X_h$ is reduced.

See also

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References

• Aroca, José Manuel; Hironaka, Heisuke; Vicente, José Luis (3 November 2018). Complex Analytic Desingularization. doi:10.1007/978-4-431-49822-3. ISBN 978-4-431-49822-3.
• Bloom, Thomas; Herrera, Miguel (1969). "De Rham cohomology of an analytic space". Inventiones Mathematicae. 7 (4): 275–296. Bibcode:1969InMat...7..275B. doi:10.1007/BF01425536. S2CID 122113902.
• Cartan, H.; Bruhat, F.; Cerf, Jean; Dolbeault, P.; Frenkel, Jean; Hervé, Michel; Malatian; Serre, J-P. "Séminaire Henri Cartan, Tome 4 (1951-1952)". (no.10-13)
• Fischer, G. (14 November 2006). Complex Analytic Geometry. Springer. ISBN 978-3-540-38121-1.
• Gunning, Robert Clifford; Rossi, Hugo (2009). "Chapter III. Variety (Sec. B. Analytic covers)". Analytic Functions of Several Complex Variables. American Mathematical Soc. ISBN 9780821821657.
• Gunning, Robert Clifford; Rossi, Hugo (2009). "Chapter V. Analytic spaces". Analytic Functions of Several Complex Variables. American Mathematical Soc. ISBN 9780821821657.
• Grauert, Hans; Remmert, Reinhold (1958). "Komplexe Räume". Mathematische Annalen. 136 (3): 245–318. doi:10.1007/BF01362011. S2CID 121348794.
• Grauert, H.; Remmert, R. (6 December 2012). Coherent Analytic Sheaves. Springer. ISBN 978-3-642-69582-7.
• Grauert, H.; Peternell, Thomas; Remmert, R. (9 March 2013). Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis. Springer. ISBN 978-3-662-09873-8.
• Grothendieck, Alexander; Raynaud, Michèle (2002). "Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique". Revêtements étales et groupe fondamental (SGA 1) (in français). arXiv:math/0206203. doi:10.1007/BFb0058656. ISBN 978-2-85629-141-2.
• Hartshorne, Robin (1970). Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics. Vol. 156. doi:10.1007/BFb0067839. ISBN 978-3-540-05184-8.
• Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. MR 0463157. S2CID 197660097. Zbl 0367.14001.
• Huckleberry, Alan (2013). "Hans Grauert (1930–2011)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 115: 21–45. arXiv:1303.6933. doi:10.1365/s13291-013-0061-7. S2CID 119685542.
• Remmert, Reinhold (1998). "From Riemann Surfaces to Complex Spaces". Séminaires et Congrès. Zbl 1044.01520.
• Serre, Jean-Pierre (1956). "Géométrie algébrique et géométrie analytique". Annales de l'Institut Fourier. 6: 1–42. doi:10.5802/aif.59. ISSN 0373-0956. MR 0082175.
• Tognoli, A. (2 June 2011). Tognoli, A (ed.). Singularities of Analytic Spaces: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Bressanone (Bolzano), Italy, June 16-25, 1974. doi:10.1007/978-3-642-10944-7. ISBN 978-3-642-10944-7.
• "Chapter II. Preliminaries". Zariski-decomposition and Abundance. Mathematical Society of Japan Memoirs. Vol. 14. Mathematical Society of Japan. 2004. pp. 13–78. doi:10.2969/msjmemoirs/01401C020. ISBN 978-4-931469-31-0.
• Flores, Arturo Giles; Teissier, Bernard (2018). "Local polar varieties in the geometric study of singularities". Annales de la Faculté des Sciences de Toulouse: Mathématiques. 27 (4): 679–775. arXiv:1607.07979. doi:10.5802/afst.1582. S2CID 119150240.

Future reading

• Huckleberry, Alan (2013). "Hans Grauert (1930–2011)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 115: 21–45. arXiv:1303.6933. doi:10.1365/s13291-013-0061-7. S2CID 256084531.

External links

• Kiran Kedlaya. 18.726 Algebraic Geometry (LEC # 30 - 33 GAGA)Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA.
• Tasty Bits of Several Complex Variables (p. 137) open source book by Jiří Lebl BY-NC-SA.