Chern ClassEdit
Chern classes are a family of topological invariants attached to complex vector bundles. They live in the cohomology of the base space and encode how a bundle twists and turns over that space. First introduced in the work of Shiing-Shen Chern in the mid-20th century, these classes provide a bridge between differential geometry and topology, turning geometric data such as connections and curvature into global, algebraic obstructions. In a precise sense, they form the most fundamental set of characteristic classes for complex bundles, in a way that is natural, functorial, and computable across a broad range of settings.
Conceptually, one can view the total Chern class c(E) of a complex vector bundle E as the formal sum c(E) = 1 + c1(E) + c2(E) + ..., where each ck(E) is a cohomology class of degree 2k. The classes ck(E) live in the integral cohomology group H^{2k}(X; Z) for a bundle over a space X and they are arranged so that they behave predictably under operations such as direct sums and pullbacks. The total Chern class is multiplicative under direct sums (c(E ⊕ F) = c(E) c(F)) and natural with respect to pullbacks, making Chern classes a central tool for distinguishing bundles that are not isomorphic. The construction and interpretation of these classes connect several mathematical ideas, including curvature forms from connections, algebraic geometry via the Chow ring, and the broader theory of characteristic classes characteristic class.
Header: History and overview The development of Chern classes brought together differential geometry, topology, and later algebraic geometry. Chern’s early work developed the differential-geometric approach to characteristic classes, and the later Chern–Weil theory formalized how curvature data from a connection on a bundle determines closed forms whose cohomology classes are invariant under changes of connection. For a complex vector bundle equipped with a connection, the curvature form determines the Chern forms, which represent the Chern classes in de Rham cohomology. This viewpoint ties together local geometric data with global topological invariants, and it underpins many computations across geometry and physics. For those interested in the algebraic side, Grothendieck’s development of Chern classes in the setting of algebraic geometry extended these ideas to holomorphic and algebraic vector bundles, tying them to the Chow ring and to powerful global theorems such as the Grothendieck–Riemann–Roch theorem Chern-Weil theory Grothendieck-Riemann-Roch algebraic geometry Chow ring.
Header: Construction and basic properties - Curvature and forms. Given a complex vector bundle E → X with a connection ∇, the curvature F is a 2-form on X with values in End(E). The Chern forms are constructed from invariant polynomials in F (for example, traces of powers of F). These differential forms are closed and their de Rham cohomology classes are independent of the chosen connection, representing the integral Chern classes ck(E) in H^{2k}(X; Z) after appropriate normalization. The total Chern class can be written formally as c(E) = det(I + (i/2π)F), a compact way of packaging all ck together. This curvature-based viewpoint is the essence of Chern-Weil theory.
First Chern class and line bundles. For a complex line bundle L, there is a single integer class c1(L) in H^2(X; Z) that detects the twisting of L. Line bundles are classified, up to isomorphism, by their first Chern class; conversely, given an integral class in H^2(X; Z) you can realize a line bundle whose c1 equals that class. In geometric terms, integrating c1 over a 2-cycle in X yields the degree of the line bundle over that cycle. This special case already illustrates how Chern classes translate local geometric information (the connection) into global topological data. See also line bundle.
Higher Chern classes. Beyond c1, the higher classes c2, c3, … capture higher-order twisting of a bundle. They live in higher even-degree cohomology groups, and together they determine much of the topological complexity of E. The Whitney sum formula, c(E ⊕ F) = c(E) c(F), expresses how twisting behaves under direct sums, while naturality under pullback means that mapping a space into X transfers the bundle’s Chern data along with the map. For a deeper algebraic perspective, one can study how these classes sit inside the framework of characteristic class theory and relate to other invariants such as the Stiefel–Whitney and Pontryagin classes for real bundles Stiefel-Whitney class Pontryagin class.
Examples and computations. The most familiar example is the 2-sphere S^2, where the tangent bundle TS^2 has c1(TS^2) equal to the Euler class, which is twice the generator of H^2(S^2; Z). This reflects the Gauss–Bonnet theorem and the intrinsic curvature of the sphere. In complex projective space CP^n, the hyperplane line bundle O(1) has a canonical first Chern class, and the total Chern class of the tangent bundle TCP^n has a compact expression c(TCP^n) = (1 + h)^{n+1} / (1 + (n+1)h), where h denotes the hyperplane class in H^2(CP^n; Z) S^2 CP^n.
Relation to the broader landscape of characteristic classes. Chern classes sit in the family of characteristic classes that assign cohomology classes to vector bundles in a way that is natural with respect to pullbacks and bundle operations. They are the prototypical invariants for complex bundles and are related to, but distinct from, real-characteristic classes like Stiefel–Whitney classes and Pontryagin classes. The Chern character ch(E), which lives in rational cohomology, provides a linearized enlargement of the Chern classes that is central in index theory and in the study of K-theory characteristic class K-theory.
Header: Chern classes in algebraic geometry and topology In the setting of complex algebraic geometry, Chern classes of holomorphic vector bundles on smooth varieties carry the same essential information, but they are interpreted within the algebraic framework of the Chow ring rather than singular cohomology. The splitting principle, the construction of Chern classes via Schubert calculus, and the interaction with the Chow ring enable a robust computational toolkit for questions in intersection theory and enumerative geometry. The Grothendieck–Riemann–Roch theorem, which links pushforwards in cohomology and in K-theory to characteristic classes, is a cornerstone of this viewpoint and shows how Chern classes connect geometry with algebraic and arithmetic information Grothendieck-Riemann-Roch Chow ring K-theory.
Header: Applications in physics and geometry Chern classes appear naturally in gauge theories and general relativity as measures of topological charge. The second Chern class, for example, counts instanton number in Yang–Mills theory, encoding how gauge fields wind around nontrivial cycles in spacetime. The first Chern class governs the quantization of magnetic flux in abelian gauge theories and in the study of line bundles over spaces with nontrivial topology. In condensed matter physics and quantum mechanics, Chern numbers appear as invariants of parameter spaces and carry physical consequences such as quantized conductance in the quantum Hall effect. The ubiquity of Chern classes across mathematics and physics reflects their role as universal indicators of twisting and obstruction, built directly from curvature data and expressed in global, topological terms. See also Yang-Mills theory Chern character.
Header: Generalizations and singular spaces While Chern classes are defined for smooth bundles over smooth spaces, there are meaningful extensions to more singular or generalized settings. For singular spaces, one uses constructions such as the Chern–Schwartz–MacPherson (CSM) classes, which extend the notion of Chern classes to a broad class of singular varieties while preserving many functorial properties. These generalized classes provide a way to ask and answer questions about topology and geometry beyond the realm of smooth manifolds, and they interact with other invariants in algebraic geometry and topology Chern-Schwartz-MacPherson.
Header: See also - characteristic class - vector bundle - line bundle - Chern–Weil theory - Chern character - K-theory - Chow ring - Grothendieck-Riemann-Roch - Chow ring