Chern class
In algebraic topology and differential geometry, Chern classes are characteristic classes c_k(E) \in H^{2k}(X; \mathbb{Z}) defined for a complex vector bundle E of rank n over a topological space X, providing integer cohomology invariants that quantify the bundle's deviation from triviality.[1] Introduced by Shiing-Shen Chern in his 1946 paper on Hermitian manifolds, these classes generalize earlier notions like the Euler class and form the foundation for studying vector bundles via their total Chern class c(E) = 1 + c_1(E) + \cdots + c_n(E).[2] They satisfy key axioms including naturality under bundle pullbacks, multiplicativity for Whitney sums c(E \oplus F) = c(E) \cup c(F), and normalization on the canonical line bundle over complex projective space, where c_1 generates H^2(\mathbb{CP}^\infty; \mathbb{Z}).[3] The first Chern class c_1(E) corresponds to the Euler class of the underlying real bundle and detects line bundles, while higher classes c_k for k \geq 2 capture more intricate obstructions to triviality, with all c_k = 0 for k > n.[1] In differential geometry, Chern classes arise via the Chern-Weil homomorphism, which associates them to the curvature of a connection on the bundle, yielding closed differential forms whose cohomology classes are independent of the connection chosen.[2] This bridges topology and geometry, enabling applications such as the Chern-Gauss-Bonnet theorem, which equates the integral of the top Chern class to the Euler characteristic of a manifold.[4] Chern classes play a central role in algebraic geometry through Grothendieck's axiomatic formulation, where they classify holomorphic vector bundles on projective varieties and appear in the Hirzebruch-Riemann-Roch theorem for computing dimensions of cohomology spaces.[5] In broader contexts, they relate to other characteristic classes like Pontryagin classes via p_k = (-1)^k c_{2k} for real bundles viewed as complex,[6] and they underpin index theory and K-theory, influencing modern developments in string theory and quantum field theory.[3] Their universality stems from the fact that every complex vector bundle is classified by a map to the Grassmannian, whose cohomology ring is generated by the Chern classes of the tautological bundle.[3]Fundamentals
Basic Idea and Motivation
Chern classes were introduced by Shiing-Shen Chern in 1946 as characteristic classes for Hermitian manifolds, providing a means to generalize the Euler class from oriented real vector bundles to the complex setting.[2] This development addressed the need to capture topological invariants of complex structures, where the Euler class alone proved insufficient for describing the full range of obstructions in higher-dimensional complex bundles.[7] Intuitively, Chern classes serve as topological measures of the "twisting" or non-triviality in complex vector bundles, linking local holomorphic or geometric data to global topological features. They detect obstructions to the existence of non-vanishing sections, much like how the Euler class identifies zeros of generic sections in oriented real bundles, but extended to encode the complex linear algebra underlying the bundle's fibers.[7] For instance, the top Chern class of a complex bundle coincides with the Euler class of its underlying real oriented bundle, offering a direct analogy to the Euler characteristic, which quantifies orientability and section zeros on manifolds.[7] A key motivation arises from differential geometry, where Chern classes preview the role of connections on bundles: they can be represented by closed differential forms derived from the curvature of such connections, bridging local metric properties with topological invariants without relying on explicit computations.[7] This connection highlights their utility in studying how infinitesimal geometric data, like curvature, aggregates to global topological obstructions in complex geometries.[2] As the simplest case, line bundles illustrate this by having a single non-trivial Chern class that classifies their isomorphism types topologically.[7]Chern Classes of Line Bundles
The first Chern class c_1(L) of a complex line bundle L over a smooth manifold X is the cohomology class in H^2(X, \mathbb{Z}) represented by \frac{1}{2\pi i} d \log s, where s is a local section of L. This expression arises from the Čech-de Rham cohomology associated to the transition functions of L, viewed as a principal U(1)-bundle. The class c_1(L) is independent of the choice of local section s or trivialization of L, as changes in section correspond to multiplication by a nowhere-vanishing function, whose logarithm contributes an exact form to the cohomology class. Equivalently, in topological terms, c_1(L) equals the Euler class of the underlying oriented real 2-plane bundle L_\mathbb{R}.[8] For a holomorphic line bundle L over a complex manifold X, the first Chern class satisfies c_1(L) = [\operatorname{div}(s)], where s is a meromorphic section of L and [\operatorname{div}(s)] denotes the cohomology class of its divisor (zeros minus poles).[9] The trivial line bundle has c_1 = 0, as it admits a global nowhere-vanishing section with vanishing divisor.[8] For the tautological line bundle \gamma on \mathbb{CP}^1, c_1(\gamma) = -H, where H is the positive generator of H^2(\mathbb{CP}^1, \mathbb{Z}) (the hyperplane class).[8] Since c_1(L) \in H^2(X, \mathbb{Z}), its pairing with any 2-cycle in X yields an integer.[8]Constructions
Chern–Weil Theory
The Chern–Weil theory constructs the Chern classes of a complex vector bundle through the geometry of connections and their curvatures. For a smooth complex vector bundle E \to M of rank r over a smooth manifold M, let \nabla be a connection on E. In a local trivialization, \nabla is represented by a \mathfrak{u}(r)-valued 1-form A, and the curvature form is the \mathfrak{u}(r)-valued 2-form \Omega = dA + A \wedge A \in \Omega^2(M, \mathfrak{u}(r)). This curvature measures the failure of \nabla to be flat and lies in the space of endomorphism-valued 2-forms globally.[10][11] The total Chern form associated to (E, \nabla) is defined as the determinant c(E, \nabla) = \det\left( I + \frac{i}{2\pi} \Omega \right) = 1 + c_1(E, \nabla) + \cdots + c_r(E, \nabla), where I denotes the identity endomorphism, and each component c_k(E, \nabla) is a closed $2k-form on M. These components arise from the expansion of the determinant in terms of the eigenvalues of \frac{i}{2\pi} \Omega, using elementary symmetric polynomials. Equivalently, the Chern forms can be represented using traces of powers of the curvature, where the k-th Chern form involves terms like \operatorname{Tr}(\Omega^k), adjusted via Newton identities to match the symmetric polynomial structure.[10][11] The de Rham cohomology classes c_k(E) = [c_k(E, \nabla)] \in H^{2k}_{\mathrm{dR}}(M, \mathbb{R}) are independent of the choice of connection \nabla, as the difference c(E, \nabla_1) - c(E, \nabla_2) is an exact form for any two connections \nabla_1 and \nabla_2 on E. This invariance follows from the fact that the curvature difference corresponds to the Maurer–Cartan structure equation, making the Chern classes topological invariants of the bundle. The normalization by the factor \frac{i}{2\pi} ensures that these classes are integral, lying in the image of the map H^{2k}(M, \mathbb{Z}) \to H^{2k}_{\mathrm{dR}}(M, \mathbb{R}).[10][11]Axiomatic Approaches
One of the foundational ways to define Chern classes is through an axiomatic characterization that specifies their behavior as natural transformations from the category of complex vector bundles to cohomology groups. In the classical topological setting, Friedrich Hirzebruch provided such a characterization for complex vector bundles over paracompact Hausdorff spaces X equipped with integer cohomology H^*(X; \mathbb{Z}).[12] The total Chern class c(E) = 1 + c_1(E) + \cdots + c_r(E) \in H^*(X; \mathbb{Z}), where r = \rank(E), is required to satisfy three key axioms:- Naturality: For any continuous map f: Y \to X, the induced map on cohomology pulls back the Chern classes compatibly, i.e., f^* c(E) = c(f^* E).
- Whitney sum formula: The Chern class is multiplicative under direct sums, c(E \oplus F) = c(E) \cup c(F), where \cup denotes the cup product in cohomology.
- Normalization: The Chern class of the trivial bundle \epsilon^n of rank n is the unit c(\epsilon^n) = 1, and for the tautological line bundle \mathcal{O}(-1) over \mathbb{CP}^\infty, c_1(\mathcal{O}(-1)) generates H^2(\mathbb{CP}^\infty; \mathbb{Z}) negatively.[12]
Properties
General Properties
The total Chern class of a complex vector bundle E over a topological space X is defined as c(E) = 1 + c_1(E) + c_2(E) + \cdots + c_r(E) \in H^*(X; \mathbb{Z}), where r is the rank of E and c_k(E) = 0 for k > r.[15] This total class is multiplicative under direct sums: for bundles E and F over X, c(E \oplus F) = c(E) \cup c(F).[15] The individual Chern classes satisfy the Whitney sum formula, c_k(E \oplus F) = \sum_{i=0}^k c_i(E) \cup c_{k-i}(F), with c_0 = 1.[3] Chern classes exhibit functoriality with respect to bundle maps and pullbacks. If f: Y \to X is a continuous map and E is a complex vector bundle over X, then the pullback bundle f^*E over Y satisfies f^* c(E) = c(f^*E), meaning each c_k(f^*E) = f^* c_k(E).[16] This naturality ensures that Chern classes are well-defined characteristic classes compatible with base change.[17] The splitting principle provides a powerful tool for computations: for any complex vector bundle E of rank r over X, there exists a map g: Z \to X to a flag manifold Z (such as a product of projective spaces) such that g^*E splits as a direct sum of line bundles L_1 \oplus \cdots \oplus L_r over Z, and g^* c(E) = \prod_{i=1}^r (1 + c_1(L_i)).[15] Consequently, the total Chern class of E can be expressed formally as c(E) = \prod_{i=1}^r (1 + x_i), where the x_i are the formal Chern roots satisfying the symmetric polynomial relations for the elementary symmetric functions in the c_k(E).[3] This virtual splitting reduces general properties to those of line bundles without altering the ring structure of the cohomology.[18]Top Chern Class
For a complex vector bundle E of rank n over a space X, the top Chern class c_n(E) is the unique component in H^{2n}(X; \mathbb{Z}) of the total Chern class c(E) = 1 + c_1(E) + \cdots + c_n(E).[15] This class coincides with the Euler class e(E) of the underlying oriented real vector bundle of rank $2n, up to sign convention, providing a direct link between complex and oriented real characteristic classes.[8] The identification c_n(E) = e(E) follows from the naturality of both classes and their agreement on line bundles, extended via the splitting principle.[15] Geometrically, c_n(E) represents the Poincaré dual of the homology class of the zero locus of a generic section of E, assuming transversality to the zero section; this locus is a closed submanifold of codimension $2n whose fundamental class pairs with cycles to yield intersection numbers determined by c_n(E).[8] For the tangent bundle TM of a compact complex n-manifold M, the pairing \langle c_n(TM), [M] \rangle equals the Euler characteristic \chi(M), reflecting the topological invariant via the index of the zero set of a generic holomorphic vector field.[19] This integrality arises from the cohomological definition and the fact that \chi(M) counts signed zeros of sections, consistent with Poincaré duality.[19] The top Chern class vanishes, c_n(E) = 0, if and only if E admits a nowhere-zero section, as such a section trivializes the obstruction class in the cohomology group.[8] In terms of other characteristic classes, the Segre classes s_k(E) are defined via the formal inverse of the total Chern class, s(E) = 1 / c(E) = \sum (-1)^k s_k(E), so that higher Segre classes incorporate c_n(E) in their expansion through multiplicative relations in the cohomology ring.[15] The Whitney sum formula extends to the top class by setting c_k = 0 for k > n, yielding c_n(E \oplus F) = c_n(E) + c_{n-1}(E) c_1(F) + \cdots + c_1(E) c_{n-1}(F) + c_n(F) for compatible ranks.[15]Examples
Complex Projective Spaces
The tangent bundle T\mathbb{CP}^n of the complex projective space \mathbb{CP}^n provides a fundamental example for computing Chern classes, as it arises from a short exact sequence of holomorphic vector bundles known as the Euler sequence:$0 \to \mathcal{O}_{\mathbb{CP}^n} \to \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus (n+1)} \to T\mathbb{CP}^n \to 0.
Here, \mathcal{O}_{\mathbb{CP}^n} denotes the trivial line bundle, and \mathcal{O}_{\mathbb{CP}^n}(1) is the hyperplane line bundle (dual to the tautological line bundle) with first Chern class h = c_1(\mathcal{O}_{\mathbb{CP}^n}(1)), the positive generator of H^2(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}.[20][21] Since the Euler sequence is an extension of vector bundles, the total Chern class of the tangent bundle is determined by the multiplicativity of Chern classes: c(T\mathbb{CP}^n) = c(\mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus (n+1)}) / c(\mathcal{O}_{\mathbb{CP}^n}). The trivial bundle has total Chern class 1, while the direct sum of n+1 copies of \mathcal{O}_{\mathbb{CP}^n}(1) has total Chern class (1 + h)^{n+1} by the Whitney sum formula. Thus,
c(T\mathbb{CP}^n) = (1 + h)^{n+1}.
This formula holds in the cohomology ring H^*(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} / (h^{n+1}).[8][20] Expanding the binomial, the individual Chern classes are
c_k(T\mathbb{CP}^n) = \binom{n+1}{k} h^k
for $1 \leq k \leq n, with c_0 = 1 and c_k = 0 for k > n. In particular, the first Chern class is c_1(T\mathbb{CP}^n) = (n+1) h, reflecting the degree of the bundle, and the top Chern class is c_n(T\mathbb{CP}^n) = (n+1) h^n.[8] The integral of the top Chern class over \mathbb{CP}^n gives the Euler characteristic:
\int_{\mathbb{CP}^n} c_n(T\mathbb{CP}^n) = (n+1) \int_{\mathbb{CP}^n} h^n = n+1,
since the fundamental class satisfies \int_{\mathbb{CP}^n} h^n = 1. This result aligns with the topological Euler characteristic of \mathbb{CP}^n, computed via its cell decomposition or otherwise as \chi(\mathbb{CP}^n) = n+1.[8][20] This computation exemplifies the Chern-Gauss-Bonnet theorem, which equates the Euler characteristic of an even-dimensional oriented Riemannian manifold to the integral of a characteristic form built from the curvature; for complex manifolds like \mathbb{CP}^n, the theorem identifies this form with the top Chern class of the tangent bundle. Shiing-Shen Chern provided an intrinsic proof of this generalization in 1945, linking local differential geometry to global topology without reference to an embedding.[22]
Hypersurfaces in Projective Space
Hypersurfaces in projective space provide concrete examples for computing Chern classes of tangent bundles using exact sequences from embedding theory. For a smooth hypersurface X \subset \mathbb{CP}^n defined by a degree d homogeneous polynomial, the tangent bundle TX fits into the short exact sequence of the normal bundle:$0 \to TX \to T\mathbb{CP}^n|_X \to \mathcal{O}_X(d) \to 0.
This sequence arises from the adjunction formula in the embedding, where \mathcal{O}_X(d) is the normal line bundle to X in \mathbb{CP}^n.[23] The total Chern class of TX follows multiplicatively from the Whitney sum formula applied to the sequence:
c(TX) = \frac{c(T\mathbb{CP}^n|_X)}{c(\mathcal{O}_X(d))},
where c(T\mathbb{CP}^n|_X) = (1 + h)^{n+1} with h = c_1(\mathcal{O}_{\mathbb{CP}^n}(1)|_X) the restricted hyperplane class, and c(\mathcal{O}_X(d)) = 1 + d h. Thus,
c(TX) = \frac{(1 + h)^{n+1}}{1 + d h}.
This formal power series expansion in h yields the individual Chern classes c_k(TX) as coefficients up to the dimension of X.[23] A prominent example is the smooth quintic threefold, the hypersurface X \subset \mathbb{CP}^4 of degree d=5 (so n=4), which is a Calabi–Yau manifold with c_1(TX) = 0. The total Chern class is
c(TX) = \frac{(1 + h)^5}{1 + 5h} = 1 + 10 h^2 - 40 h^3 + \ higher\ terms,
so the top Chern class is c_3(TX) = -40 h^3. The topological Euler characteristic is then \chi(X) = \int_X c_3(TX) = -40 \int_X h^3 = -40 \cdot 5 = -200, since \int_X h^3 = d = 5 is the degree of X.[23][24] For general smooth degree d hypersurfaces in \mathbb{CP}^n, the Chern classes c_k(TX) are the degree-k coefficients in the expansion of \frac{(1 + h)^{n+1}}{1 + d h}, with c_1(TX) = (n+1 - d) h. These computations underpin applications in enumerative geometry, such as determining genus constraints or curve counts on hypersurfaces via Hirzebruch–Riemann–Roch.[23][25] The above assumes X is smooth, requiring transverse zeros of the defining polynomial. For singular hypersurfaces, the Chern–Schwartz–MacPherson class provides a corrective extension of the tangent Chern class, incorporating terms from the singular locus via the \mu-class to ensure proper transformation under embeddings.[26]