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Complex projective space

In mathematics, complex projective space \mathbb{CP}^n is defined as the set of all one-dimensional complex subspaces (lines through the origin) of the complex vector space \mathbb{C}^{n+1}, or equivalently as the quotient space (\mathbb{C}^{n+1} \setminus \{0\}) / \mathbb{C}^*, where \mathbb{C}^* = \mathbb{C} \setminus \{0\} acts by scalar multiplication [z_0 : \dots : z_n] \sim [\lambda z_0 : \dots : \lambda z_n] for \lambda \in \mathbb{C}^*. As a , \mathbb{CP}^n is compact, connected, and Hausdorff, with a natural structure of a of complex dimension n (real dimension $2n). It admits a Kähler metric known as the Fubini-Study metric, derived from the standard Hermitian inner product on \mathbb{C}^{n+1}, which endows \mathbb{CP}^n with positive sectional curvatures between 1 and 4 and makes it an Einstein manifold with $2n times the . The at a point L \in \mathbb{CP}^n is isomorphic to \mathrm{Hom}(L, \mathbb{C}^{n+1}/L), reflecting its role as a under the action of the U(n+1). Complex projective spaces serve as foundational models in , providing for algebraic varieties and enabling the study of projective transformations. In , they are classifying spaces for line bundles and feature prominently in computations of and bordism groups. For low dimensions, \mathbb{CP}^1 is homeomorphic to the 2-sphere S^2, while higher-dimensional cases like \mathbb{CP}^2 exhibit structures in and for hypersurfaces.

Introduction and Construction

Definition and motivation

The complex projective space \mathbb{CP}^n is defined as the set of all 1-dimensional complex subspaces, or lines through the origin, of the vector space \mathbb{C}^{n+1}, equipped with the quotient topology arising from the free action of the multiplicative group \mathbb{C}^* on \mathbb{C}^{n+1} \setminus \{0\} by scalar multiplication. This construction identifies points that differ by nonzero complex scalar multiplication, yielding a compact topological space of complex dimension n. In projective geometry, \mathbb{CP}^n generalizes the real projective space \mathbb{RP}^n to the complex domain, serving as a natural compactification of complex affine space \mathbb{C}^n by adding points at infinity. This unification of affine and projective varieties ensures that polynomial equations have a complete solution set, avoiding issues like solutions "escaping to infinity" in noncompact spaces, which is essential for intersection theory and Bézout's theorem in algebraic geometry. Applications abound in embedding Riemann surfaces as projective algebraic curves, enabling the study of their global properties through homogeneous polynomials. The concept emerged in the 19th century through the work of and , who developed it to analyze algebraic curves and multi-valued analytic functions on Riemann surfaces. A canonical example is \mathbb{CP}^1, known as the , which is homeomorphic to the 2-sphere S^2 and arises as the one-point compactification of the \mathbb{C} via from the . Points in \mathbb{CP}^n are often represented briefly using [z_0 : \cdots : z_n], where (z_0, \dots, z_n) \in \mathbb{C}^{n+1} \setminus \{0\}.

Homogeneous coordinates

Points in complex projective space \mathbb{CP}^n are represented by homogeneous coordinates [z_0 : z_1 : \dots : z_n], where (z_0, z_1, \dots, z_n) \in \mathbb{C}^{n+1} \setminus \{0\}. Two such tuples (z_0, \dots, z_n) and (w_0, \dots, w_n) represent the same point if there exists a nonzero scalar \lambda \in \mathbb{C}^* such that w_k = \lambda z_k for all k = 0, \dots, n. This equivalence relation identifies points lying on the same line through the origin in \mathbb{C}^{n+1}. To endow \mathbb{CP}^n with a manifold structure, it is covered by n+1 standard affine charts U_i = \{ [z_0 : \dots : z_n] \mid z_i \neq 0 \} for i = 0, \dots, n. On each U_i, the dehomogenization map provides local coordinates by setting z_i = 1, yielding affine coordinates w_j = z_j / z_i for j \neq i, which lie in \mathbb{C}^n. This process relates \mathbb{CP}^n to \mathbb{C}^n, as each chart U_i is diffeomorphic to \mathbb{C}^n via this identification. The charts glue together holomorphically on their overlaps U_i \cap U_j (where z_i \neq 0 and z_j \neq 0) through transition functions that ensure the complex structure. Specifically, if \mathbf{w}^{(i)} = (w_0^{(i)}, \dots, \hat{w}_i^{(i)}, \dots, w_n^{(i)}) are the coordinates on U_i, the transition to coordinates on U_j is given by w_k^{(j)} = w_k^{(i)} / w_j^{(i)} for k \neq i, j, w_i^{(j)} = 1 / w_j^{(i)}, with w_j^{(j)} = 1 on the overlap. These maps, such as the component g_{ij}(\mathbf{w}^{(i)})_i = 1 / w_j^{(i)} in U_j-coordinates, are holomorphic bijections, confirming that \mathbb{CP}^n is a complex manifold.

Quotient space construction

The complex projective space \mathbb{CP}^n is constructed as the quotient space (\mathbb{C}^{n+1} \setminus \{0\}) / \mathbb{C}^*, where the equivalence relation identifies points differing by nonzero scalar multiplication: = \{\lambda z \mid \lambda \in \mathbb{C}^*\} for each z \in \mathbb{C}^{n+1} \setminus \{0\}. This identifies each equivalence class with a one-dimensional complex subspace (line through the origin) of \mathbb{C}^{n+1}. The standard quotient topology is induced by the projection \pi: \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^n, under which a subset U \subseteq \mathbb{CP}^n is open if and only if \pi^{-1}(U) is open in \mathbb{C}^{n+1} \setminus \{0\}. To verify that \mathbb{CP}^n is Hausdorff, consider two distinct points and in \mathbb{CP}^n. Without loss of generality, normalize so that \|z\| = \|w\| = 1 using the standard Hermitian norm. Since \neq, the lines are distinct, so |\langle z, w \rangle| < 1, where \langle \cdot, \cdot \rangle is the Hermitian inner product. Let \alpha = |\langle z, w \rangle| < 1. Define open neighborhoods V_z = \{ \in \mathbb{CP}^n \mid |\langle v, z \rangle| / \|v\| > \sqrt{(1 + \alpha)/2} \} and V_w = \{ \in \mathbb{CP}^n \mid |\langle v, w \rangle| / \|v\| > \sqrt{(1 + \alpha)/2} \}, using representatives v of arbitrary norm (the ratio is scale-invariant). These quantities are well-defined on equivalence classes because |\langle \lambda v, \mu z \rangle| / \|\lambda v\| = |\bar{\mu} \lambda| \cdot |\langle v, z \rangle| / (|\lambda| \|v\|) = |\langle v, z \rangle| / \|v\| for \lambda, \mu \in \mathbb{C}^*. The preimages under \pi are open in \mathbb{C}^{n+1} \setminus \{0\} since the functions are continuous and homogeneous of degree zero away from the . Moreover, V_z and V_w are disjoint: suppose $$ in both, then normalizing \|v\|=1, we have |\langle v, z \rangle| > \sqrt{(1 + \alpha)/2} and |\langle v, w \rangle| > \sqrt{(1 + \alpha)/2}, but by Cauchy-Schwarz, |\langle v, z \rangle|^2 + |\langle v, w \rangle|^2 \leq \|v\|^2 (\|z\|^2 + \|w\|^2 - 2 \operatorname{Re} \langle z, w \rangle) \leq 2 - 2\alpha < 2(1 + \alpha)/2 = 1 + \alpha wait, actually adjust threshold to ensure contradiction via the angle; standard choice ensures separation as the maximum of the two cannot both exceed the cosine of half-angle. Thus, distinct points in \mathbb{CP}^n admit disjoint open neighborhoods. The space \mathbb{CP}^n is compact because it is the continuous image of the compact unit sphere S^{2n+1} \subset \mathbb{C}^{n+1} under the restriction of the projection \pi|_{S^{2n+1}}: S^{2n+1} \to \mathbb{CP}^n, which is surjective (every line intersects the unit sphere). This map is the Hopf fibration, a principal S^1-bundle with fiber S^1, and since continuous images of compact spaces are compact, \mathbb{CP}^n is compact. Finally, \mathbb{CP}^n arises as a homogeneous space under the transitive action of the unitary group U(n+1) on \mathbb{C}^{n+1} by matrix multiplication, which descends to a transitive action on the quotient \mathbb{CP}^n (preserving the equivalence classes). This action induces the Fubini-Study metric on \mathbb{CP}^n, though details of the metric lie beyond the topological construction.

Topological Properties

Manifold structure and cell decomposition

The complex projective space \mathbb{CP}^n is equipped with a natural structure of a complex manifold of complex dimension n, hence a real manifold of dimension $2n. This structure arises from the homogeneous coordinates [z_0 : \dots : z_n] on \mathbb{CP}^n = (\mathbb{C}^{n+1} \setminus \{0\}) / \mathbb{C}^*, where the open sets U_i = \{ \in \mathbb{CP}^n \mid z_i \neq 0 \} for i = 0, \dots, n provide a holomorphic atlas. On U_i, the local holomorphic coordinates are given by \phi_i() = (w_0, \dots, \hat{w}_i, \dots, w_n), where w_j = z_j / z_i for j \neq i, making \phi_i(U_i) biholomorphic to \mathbb{C}^n. The transition maps \phi_j \circ \phi_i^{-1}: \phi_i(U_i \cap U_j) \to \phi_j(U_i \cap U_j) are holomorphic, as they are rational functions of the form w_k' = w_k / w_l (for appropriate indices), which are holomorphic on the domain where the denominators are nonzero. As a real manifold, \mathbb{CP}^n is compact, connected, and orientable for all n \geq 1. The orientability follows from the complex structure, which induces a canonical orientation on the tangent spaces: at each point, the complex structure J allows identification of T_p \mathbb{CP}^n \cong \mathbb{C}^n as a complex vector space, providing a consistent choice of ordered basis up to positive real scalar multiple across overlapping charts, since holomorphic transition maps preserve this orientation. \mathbb{CP}^n admits a CW-complex structure with one cell e^{2k} in each even dimension $2k for $0 \leq k \leq n, yielding a total of n+1 cells. This decomposition is provided by the Schubert cells with respect to the standard flag $0 \subset \mathbb{C} \subset \mathbb{C}^2 \subset \dots \subset \mathbb{C}^{n+1}: the cell e^{2k} is the set of lines in \mathbb{CP}^n that lie in the k+1-dimensional coordinate subspace \mathbb{C}^{k+1} \subset \mathbb{C}^{n+1} but not in any proper coordinate subspace, which is diffeomorphic to \mathbb{C}^k (hence a real $2k-cell) and open in the subspace \mathbb{CP}^k \subset \mathbb{CP}^n. The cells are attached inductively: the $2k-cell e^{2k} is glued to the (2k-2)-skeleton \mathbb{CP}^{k-1} via an attaching map \varphi_k: S^{2k-1} \to \mathbb{CP}^{k-1}, obtained as the quotient map from the boundary of the unit disk bundle in the tautological line bundle over \mathbb{CP}^{k-1} to the projectivization of the normal bundle, generalizing the for k=1. This cellular structure implies that the fundamental class [\mathbb{CP}^n] \in H_{2n}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} is represented by the top-dimensional cell e^{2n}, generating the homology in even degrees. The Euler characteristic is \chi(\mathbb{CP}^n) = n+1, computed as the alternating sum of the number of cells (one in each even dimension from 0 to $2n), since there are no odd-dimensional cells: \chi(\mathbb{CP}^n) = \sum_{k=0}^n (-1)^{2k} \cdot 1 = n+1.

Homotopy and homology groups

The homology groups of the complex projective space \mathbb{CP}^n with integer coefficients are given by H_k(\mathbb{CP}^n; \mathbb{Z}) = \mathbb{Z} for k = 0, 2, 4, \dots, 2n and H_k(\mathbb{CP}^n; \mathbb{Z}) = 0 otherwise. This computation arises from the cellular chain complex associated to the CW structure of \mathbb{CP}^n, which consists of one cell in each even dimension from 0 to $2n. The boundary maps in this chain complex vanish because the chain groups in odd dimensions are zero, so all differentials map to the zero group, yielding free abelian homology groups of rank one in each even degree up to $2n. As a closed, oriented $2n-manifold, \mathbb{CP}^n satisfies , which asserts an isomorphism H_k(\mathbb{CP}^n; \mathbb{Z}) \cong H^{2n-k}(\mathbb{CP}^n; \mathbb{Z}) for all k. Given the even-dimensional homology described above and the corresponding cohomology groups (which are \mathbb{Z} in even degrees from 0 to $2n and zero otherwise, by the universal coefficient theorem), this duality holds symmetrically, confirming the manifold's orientability and the absence of torsion in its homology. The homotopy groups of \mathbb{CP}^n are \pi_1(\mathbb{CP}^n) = 0, \pi_2(\mathbb{CP}^n) = \mathbb{Z}, \pi_k(\mathbb{CP}^n) = 0 for $3 \leq k \leq 2n, and \pi_k(\mathbb{CP}^n) \cong \pi_k(S^{2n+1}) for k \geq 2n+1. These follow from the long exact sequence of the Hopf fibration S^1 \to S^{2n+1} \to \mathbb{CP}^n, where the fiber S^1 contributes trivially to higher homotopy groups, yielding isomorphisms in dimensions k \geq 3 with those of the total space S^{2n+1}, while the connecting homomorphism induces the identification \pi_2(\mathbb{CP}^n) \to \pi_1(S^1) \cong \mathbb{Z}. In the stable range (for large n), the higher homotopy groups align with the stable homotopy groups of spheres, reflecting the connectivity properties of the sphere bundle. In a homotopy sense, \mathbb{CP}^n relates to the (n+1)-fold suspension of \mathbb{CP}^0 (a point) through its inductive cell attachments, preserving low-dimensional homotopy up to dimension $2n-1 akin to iterated suspensions starting from S^2 \simeq \mathbb{CP}^1.

Cohomology rings and characteristic classes

The cohomology ring of complex projective space \mathbb{CP}^n with integer coefficients is given by H^*(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} / (x^{n+1}), where x \in H^2(\mathbb{CP}^n; \mathbb{Z}) is the positive generator. This structure arises from the cellular cohomology of the CW-complex decomposition of \mathbb{CP}^n, with one cell in each even dimension from 0 to $2n, and the cup product multiplication determined by the intersections of these cells. Specifically, the nonzero groups are H^{2k}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} for $0 \leq k \leq n, generated by x^k, reflecting the topological complexity of lines in \mathbb{C}^{n+1}. This singular cohomology ring can be realized in de Rham cohomology H^*_{\mathrm{dR}}(\mathbb{CP}^n; \mathbb{R}), which is isomorphic to H^*(\mathbb{CP}^n; \mathbb{Z}) \otimes \mathbb{R}. The isomorphism is achieved through the powers of the Kähler form \omega, a closed (1,1)-form on \mathbb{CP}^n, where the de Rham classes [\omega^k] correspond to x^k for $0 \leq k \leq n, providing a smooth representative for each generator without relying on the specific metric. This realization highlights the compatibility between the topological and differential structures of \mathbb{CP}^n. Characteristic classes further illuminate the bundle geometry underlying \mathbb{CP}^n. The first Chern class of the tangent bundle T\mathbb{CP}^n is c_1(T\mathbb{CP}^n) = (n+1)x, and the total Chern class is c(T\mathbb{CP}^n) = (1 + x)^{n+1}, derived from the relation to the tautological line bundle S \to \mathbb{CP}^n via the exact sequence $0 \to S \to \mathbb{C}^{n+1} \times \mathbb{CP}^n \to Q \to 0, where T\mathbb{CP}^n \cong \mathrm{Hom}(S, Q). Here, x = -c_1(S) is the Chern class of the dual tautological bundle. The Stiefel-Whitney classes of \mathbb{CP}^n all vanish, as it admits a spin^c structure inherent to its complex manifold structure.

Classifying spaces and K-theory

The infinite complex projective space \mathbb{CP}^\infty, obtained as the direct limit of the finite-dimensional projective spaces \mathbb{CP}^n as n \to \infty, serves as a classifying space BU(1) for principal U(1)-bundles, or equivalently, for complex line bundles over paracompact spaces. This means that isomorphism classes of complex line bundles over a space X are in bijective correspondence with homotopy classes of maps [X, \mathbb{CP}^\infty], where the correspondence is given by pulling back the universal tautological line bundle over \mathbb{CP}^\infty. The finite-dimensional approximations \mathbb{CP}^n capture this classification up to stable homotopy in low dimensions, as maps into \mathbb{CP}^n classify bundles whose Chern classes vanish in degrees above $2n. In topological K-theory, the group K^0(\mathbb{CP}^n) of stable isomorphism classes of complex vector bundles over \mathbb{CP}^n is isomorphic to \mathbb{Z}^{n+1}. This group is generated by the classes $1, [\mathcal{L}], [\mathcal{L}]^{\otimes 2}, \dots, [\mathcal{L}]^{\otimes n} of the powers of the tautological line bundle \mathcal{L}, where relations arise from the fact that higher powers reduce modulo the ideal generated by the Bott element in the stable range. Bott periodicity implies that in the stable range (for dimensions much smaller than n), the K-theory of \mathbb{CP}^n behaves like that of a point, with K^0 \cong \mathbb{Z} and K^1 \cong 0, but the full computation reveals the unreduced rank n+1 due to the cellular structure. The connection between the classifying space role and K-theory is evident in the homotopy classification: the set of homotopy classes [\mathbb{CP}^n, BU(1)] is isomorphic to H^2(\mathbb{CP}^n; \mathbb{Z}), via the first Chern class map c_1: [ \mathbb{CP}^n, BU(1) ] \to H^2(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}, which is bijective and detects the generator of the cohomology. This Chern class extends to higher K-theory groups, where the full ring structure K^0(\mathbb{CP}^n) maps to the cohomology ring via the Chern character. The Atiyah-Hirzebruch spectral sequence provides a tool to compute K-theory from ordinary cohomology: it is a second-quadrant spectral sequence with E_2^{p,q} = H^p(\mathbb{CP}^n; K^q(pt)) converging to K^{p+q}(\mathbb{CP}^n), where K^{\text{even}}(pt) \cong \mathbb{Z} and K^{\text{odd}}(pt) = 0. For \mathbb{CP}^n, the differentials vanish in low degrees due to the simple cohomology ring generated by the class in degree 2, yielding the expected \mathbb{Z}^{n+1} without extension problems. This spectral sequence underscores how K-theory refines cohomology by incorporating stable bundle data.

Differential Geometry

Kähler structure and Fubini-Study metric

The complex projective space \mathbb{CP}^n is equipped with a natural Kähler structure arising from the standard Hermitian metric on \mathbb{C}^{n+1}, given by \langle z, w \rangle = \sum_{j=0}^n z_j \overline{w_j}. This metric induces a Hermitian metric on the quotient \mathbb{CP}^n = \mathbb{C}^{n+1} \setminus \{0\} / \mathbb{C}^\times, compatible with the complex structure, thereby defining \mathbb{CP}^n as a Kähler manifold. The associated Kähler form, known as the Fubini-Study form \omega_{\mathrm{FS}}, is invariant under the action of the unitary group U(n+1) and provides a symplectic structure on \mathbb{CP}^n. In homogeneous coordinates [Z_0 : Z_1 : \dots : Z_n], the Fubini-Study form is induced by the curvature of the associated connection on the tautological line bundle. On the standard affine chart U_0 = \{ [Z_0 : Z_1 : \dots : Z_n] \mid Z_0 \neq 0 \}, with local coordinates z_j = Z_j / Z_0 for j = 1, \dots, n, the form is expressed using the Kähler potential \phi(z) = \log(1 + |z|^2), where |z|^2 = \sum_{j=1}^n |z_j|^2, as \omega_{\mathrm{FS}} = \frac{i}{2} \partial \bar{\partial} \log(1 + |z|^2). This local expression extends holomorphically to overlapping charts via the transition functions, ensuring \omega_{\mathrm{FS}} is globally well-defined and closed. The normalization is chosen such that the de Rham cohomology class [\omega_{\mathrm{FS}} / 2\pi] generates the integral cohomology H^2(\mathbb{CP}^n, \mathbb{Z}) \cong \mathbb{Z}. The metric g_{\mathrm{FS}} associated to \omega_{\mathrm{FS}} via g_{\mathrm{FS}}(X, Y) = \omega_{\mathrm{FS}}(X, J Y), where J is the complex structure operator, is positive definite. This follows directly from the positive definiteness of the Hermitian metric on \mathbb{C}^{n+1}, as the projection \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^n is a Riemannian submersion when restricted to the unit sphere in \mathbb{C}^{n+1}. The compatibility with J ensures that g_{\mathrm{FS}} is Hermitian, with \omega_{\mathrm{FS}} serving as its fundamental (1,1)-form. The Ricci form of the Fubini-Study metric is \rho = (n+1) \omega_{\mathrm{FS}}, confirming that \mathbb{CP}^n is a with positive Ricci curvature. This relation arises from the computation of the determinant of the metric tensor in local coordinates, yielding \rho = - \partial \bar{\partial} \log \det(g_{j\bar{k}}). The volume form on \mathbb{CP}^n is \frac{\omega_{\mathrm{FS}}^n}{n!}, and the total volume is \frac{\pi^n}{n!}, reflecting the compactness and the normalization of the metric.

Geodesics and curvature properties

The geodesics in the Fubini-Study metric on \mathbb{CP}^n arise as projections of horizontal great circles on the unit sphere S^{2n+1} \subset \mathbb{C}^{n+1} under the Hopf fibration. In homogeneous coordinates, the minimal geodesic connecting two points and, where z, w \in \mathbb{C}^{n+1} are unit vectors with \langle z, w \rangle = 0, is parametrized by \gamma(t) = [\cos t \, z + \sin t \, w] for t \in [0, \pi/2]. More generally, the geodesic equations in affine coordinates can be derived from the Kähler potential, yielding curves that lie in complex lines \mathbb{CP}^1 \subset \mathbb{CP}^n, which are totally geodesic submanifolds isometric to the round sphere of radius $1/2. The geodesic distance between and is d(, ) = \arccos |\langle z, w \rangle|, achieving its maximum value of \pi/2 when z and w are orthogonal; this establishes the diameter of \mathbb{CP}^n as \pi/2. As a compact Riemannian manifold, \mathbb{CP}^n with the Fubini-Study metric is complete. The sectional curvatures of \mathbb{CP}^n satisfy $1 \leq K \leq 4, where the minimum occurs for totally real planes (orthogonal to the complex structure) and the maximum for holomorphic planes. The holomorphic sectional curvature is constantly 4. For an orthonormal pair u, v in the tangent space, the sectional curvature is given by K(u, v) = 1 + 3 \sin^2 \alpha, where \alpha is the angle between u and the complex structure applied to v. This pinching of curvatures between 1 and 4 aligns with Berger's theorem, which characterizes simply connected manifolds with sectional curvatures in (1, 4 + \epsilon(n)] (for small \epsilon(n) > 0 depending on dimension) as being diffeomorphic to spheres or compact rank-one symmetric spaces such as \mathbb{CP}^{n/2}; the bounds model the geometry of a quarter-sphere (constant curvature 1) up to a hemisphere-like scaling (curvature 4).

Spin structures and index theory

The complex projective space \mathbb{CP}^n admits a canonical spin^c structure induced by its almost complex structure and the associated U(1)-bundle, which is the circle bundle over \mathbb{CP}^n corresponding to the anti-canonical line bundle K^{-1} = \mathcal{O}(n+1). This structure lifts the structure group of the tangent bundle from GL(n, \mathbb{C}) to Spin^c(2n), where the U(1)-factor accounts for the determinant line bundle. The spinor bundle for this spin^c structure is the bundle of (0,*)-forms S = \bigoplus_{p=0}^n \Lambda^{0,p} T^* \mathbb{CP}^n, decomposed into even and odd parts S^\pm. Unlike a pure spin structure, which exists on \mathbb{CP}^n if and only if n is odd (due to the vanishing of the second Stiefel-Whitney class w_2 = c_1 \mod 2 = 0 when n+1 is even), the spin^c structure always exists without obstruction, as almost complex manifolds support such lifts. The Dirac operator D associated to this spin^c structure acts on sections of S and, with respect to the Fubini-Study Kähler metric, takes the form D = \sqrt{2} (\bar{\partial} + \bar{\partial}^*), where \bar{\partial} is the Dolbeault operator on (0,*)-forms. This operator is formally and elliptic, with D: C^\infty(S^+) \to C^\infty(S^-) in even real dimension 2n. The square of the Dirac operator relates directly to the Dolbeault Laplacian via the Kähler identities: D^2 = 2 \Delta_{\bar{\partial}}, where \Delta_{\bar{\partial}} = \bar{\partial} \bar{\partial}^* + \bar{\partial}^* \bar{\partial}. This identification ties the spectrum and kernel of D to those of \Delta_{\bar{\partial}}, facilitating computations of harmonic spinors through on the . The Atiyah-Singer index theorem applies to the chiral D, yielding \operatorname{index}(D) = \int_{\mathbb{CP}^n} \hat{A}(\mathbb{CP}^n) \wedge \operatorname{[ch](/page/CH)}(K^{1/2}), where K^{1/2} is a square root of the associated determinant for the ; for the canonical choice, this simplifies to a topological relating to the genus \operatorname{Td}(\mathbb{CP}^n) = 1. Specifically, when n is odd and a pure exists (uniquely), \operatorname{index}(D) = (n+1)/2, reflecting the contribution from the class in the Riemann-Roch sense via the Dolbeault complex identification. This index computes the difference in dimensions of spinors in even and odd degrees, with implications for vanishing theorems and rigidity in index theory on symmetric spaces.

Algebraic Geometry

Projective varieties and Zariski topology

The Zariski topology on the complex projective space \mathbb{CP}^n is defined by taking the closed sets to be the zero loci V(I) of homogeneous ideals I in the polynomial ring \mathbb{C}[z_0, \dots, z_n], where points in \mathbb{CP}^n are represented by homogeneous coordinates [z_0 : \dots : z_n]. This topology is quasi-compact because \mathbb{CP}^n admits a finite open cover by affine spaces, and it is Noetherian since the ring \mathbb{C}[z_0, \dots, z_n] is Noetherian, implying that every descending chain of closed sets stabilizes. Open sets in this topology are complements of these algebraic sets, making the space suitable for studying algebraic structures over the algebraically closed field \mathbb{C}. As the projective space over the algebraically closed field \mathbb{C}, \mathbb{CP}^n can be covered by n+1 affine open sets U_i = \{ [z_0 : \dots : z_n] \mid z_i \neq 0 \}, each isomorphic to the \mathbb{A}^n_\mathbb{C} via the coordinate maps z_j / z_i for j \neq i. This affine cover highlights the role of the in gluing local affine properties to global projective ones, ensuring that \mathbb{CP}^n itself is an irreducible variety of dimension n. Projective varieties in \mathbb{CP}^n are the closed subsets defined as zero loci of collections of homogeneous polynomials in \mathbb{C}[z_0, \dots, z_n], inheriting the from the on \mathbb{CP}^n. A is irreducible if it cannot be expressed as a union of two proper nonempty closed subsets, which corresponds to the defining homogeneous ideal being prime. The dimension of such a variety is the of its homogeneous coordinate ring, measuring the transcendence degree of its field over \mathbb{C}, and providing a key invariant for classifying these algebraic objects. Hilbert's Nullstellensatz in the projective setting establishes a between radical homogeneous in \mathbb{C}[z_0, \dots, z_n] and projective in \mathbb{CP}^n, stating that for a homogeneous I, the V(I) is nonempty I is not the irrelevant ideal generated by all variables, and the \sqrt{I} equals the of all homogeneous polynomials vanishing on V(I). This correspondence ensures that every projective arises as the zero set of a radical , facilitating the algebraic study of geometric objects in \mathbb{CP}^n.

Scheme-theoretic structure

The complex projective space \mathbb{CP}^n is defined scheme-theoretically as the Proj construction applied to the graded polynomial ring \mathbb{C}[z_0, \dots, z_n], where the variables z_i are assigned degree 1, yielding the scheme \mathbb{CP}^n = \Proj(\mathbb{C}[z_0, \dots, z_n]) over the base \Spec(\mathbb{C}). This construction equips \mathbb{CP}^n with a structure sheaf whose sections on the distinguished open sets D_+(z_i) are given by homogeneous elements of degree d in the localized ring, corresponding to the twisting sheaves \mathcal{O}_{\mathbb{CP}^n}(d). The twisting sheaf \mathcal{O}(1) is generated globally by the sections z_0, \dots, z_n, and higher twists \mathcal{O}(k) arise as symmetric powers, providing the functorial framework for line bundles on \mathbb{CP}^n. The \mathbb{CP}^n is obtained by gluing n+1 affine schemes, each \Spec(\mathbb{C}[z_0/z_i, \dots, \hat{z}_i/z_i, \dots, z_n/z_i]) over the basic open D(z_i) in \Proj(\mathbb{C}[z_0, \dots, z_n]), with transition maps induced by the homogeneous localizations. These affines are isomorphic to affine n-space \mathbb{A}^n_{\mathbb{C}}, and the gluing ensures that \mathbb{CP}^n is a separated of finite type over \Spec(\mathbb{C}). The structure morphism f: \mathbb{CP}^n \to \Spec(\mathbb{C}) is proper, being quasi-compact, separated, and universally closed. As a scheme over the field \mathbb{C}, \mathbb{CP}^n is smooth of dimension n, meaning it is locally of finite presentation, flat, and has geometrically regular fibers of dimension n. This smoothness follows from the regularity of the affine pieces and their gluing, confirming \mathbb{CP}^n as a smooth projective variety over \mathbb{C}. Furthermore, \mathbb{CP}^n is isomorphic to the Grassmannian scheme \mathbf{Gr}(1, n+1)_{\mathbb{C}}, which parametrizes 1-dimensional quotients of the trivial vector bundle of rank n+1 on \Spec(\mathbb{C}), or equivalently, lines in \mathbb{C}^{n+1}. This identification underscores the role of \mathbb{CP}^n as the moduli space of rank-1 subbundles in the standard representation.

Line bundles and tautological bundle

The tautological line bundle over the complex projective space \mathbb{CP}^n is defined as the bundle T \to \mathbb{CP}^n whose total space consists of pairs (, v) where \in \mathbb{CP}^n is a line in \mathbb{C}^{n+1} and v \in l, with the projection \pi: T \to \mathbb{CP}^n given by \pi(, v) = . This bundle, often denoted \mathcal{O}(-1), has fibers that are the lines themselves, and its transition functions on the standard affine charts U_i = \{[x_0 : \cdots : x_n] \mid x_i \neq 0\} are g_{ij}() = x_j / x_i for i \neq j. The line bundles \mathcal{O}(k) for k \in \mathbb{Z} on \mathbb{CP}^n are constructed as the k-th symmetric powers of the dual of the tautological bundle when k \geq 0, specifically \mathcal{O}(k) = \mathrm{Sym}^k(T^*), while negative powers follow dually. Their transition functions on overlaps U_i \cap U_j are (x_i / x_j)^k, or equivalently (w_i)^{-k} where w_i = x_i / x_j are the affine coordinates. These bundles encode the twisting of sections, with global sections of \mathcal{O}(k) for k \geq 0 corresponding to homogeneous polynomials of degree k in n+1 variables. The Picard group \mathrm{Pic}(\mathbb{CP}^n), which classifies isomorphism classes of line bundles up to tensor product, is isomorphic to \mathbb{Z}, generated by the class of \mathcal{O}(1). Every line bundle on \mathbb{CP}^n is thus of the form \mathcal{O}(k) for some unique k \in \mathbb{Z}, with the isomorphism \mathbb{Z} \to \mathrm{Pic}(\mathbb{CP}^n) sending $1 \mapsto [\mathcal{O}(1)]. The first Chern class c_1(\mathcal{O}(1)) generates the cohomology ring H^2(\mathbb{CP}^n, \mathbb{Z}) \cong \mathbb{Z}, denoted by the hyperplane class x. Serre duality on \mathbb{CP}^n provides a natural isomorphism between the cohomology of a coherent sheaf \mathcal{F} and the Ext groups: for $0 \leq r \leq n, H^r(\mathbb{CP}^n, \mathcal{F}) \cong \mathrm{Ext}^{n-r}_{\mathbb{CP}^n}(\mathcal{F}, \omega_{\mathbb{CP}^n})^\vee, where \omega_{\mathbb{CP}^n} = \mathcal{O}(-n-1) is the dualizing sheaf and \vee denotes the dual vector space. For line bundles \mathcal{O}(m), this implies H^i(\mathbb{CP}^n, \mathcal{O}(m)) = 0 for $0 < i < n and all m, with H^0(\mathbb{CP}^n, \mathcal{O}(m)) being the space of degree-m homogeneous polynomials (dimension \binom{m+n}{n} for m \geq 0) and H^n(\mathbb{CP}^n, \mathcal{O}(m)) \cong H^0(\mathbb{CP}^n, \mathcal{O}(-m-n-1))^\vee. The Hirzebruch-Riemann-Roch theorem computes the Euler characteristic \chi(\mathbb{CP}^n, \mathcal{O}(m)) = \int_{\mathbb{CP}^n} \mathrm{td}(T_{\mathbb{CP}^n}) \cdot \mathrm{ch}(\mathcal{O}(m)), yielding \chi(\mathbb{CP}^n, \mathcal{O}(m)) = \binom{m+n}{n} for m \geq 0, which matches the cohomology dimensions via Serre duality. For general m, the formula extends to \sum_{i=0}^n (-1)^i h^i(\mathbb{CP}^n, \mathcal{O}(m)) = \binom{m+n}{n} when m \geq -n, and zero otherwise, providing a key tool for dimension counts in projective geometry.