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Characteristic classes of vector bundles
topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since
Chern_class
Chinese-American mathematician and poet
Chern's work, most notably the Chern–Gauss–Bonnet theorem, Chern–Simons theory, and Chern classes, are still highly influential in current research in mathematics
Shiing-Shen_Chern
Mathematical theory
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal
Chern–Weil_homomorphism
Generalizations of codimension-1 subvarieties of algebraic varieties
{Pic} (X)\to \operatorname {Cl} (X),} known as the first Chern class. The first Chern class is injective if X is normal, and it is an isomorphism if X
Divisor_(algebraic_geometry)
Association of cohomology classes to principal bundles
fundamental characteristic classes known at that time (the Stiefel–Whitney class, the Chern class, and the Pontryagin classes) were reflections of the classical
Characteristic_class
Riemannian manifold with SU(n) holonomy
conjectured that compact complex manifolds of Kähler type with vanishing first Chern class always admit Ricci-flat Kähler metrics, and Shing-Tung Yau (1978), who
Calabi–Yau_manifold
Characteristic class for real vector bundles
c_{2k}(E\otimes \mathbb {C} )} denotes the 2 k {\displaystyle 2k} -th Chern class of the complexification E ⊗ C = E ⊕ i E {\displaystyle E\otimes \mathbb
Pontryagin_class
Characteristic class in algebraic topology
bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the
Todd_class
Group in algebraic geometry
H^{2}(V,{\mathcal {O}}_{V})\to \cdots .} The first arrow is the first Chern class on the Picard group c 1 : P i c ( V ) → H 2 ( V , Z ) , {\displaystyle
Néron–Severi_group
Concept in geometry
In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles as opposed to a single
Localized_Chern_class
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that
Chern–Gauss–Bonnet_theorem
Branch of algebraic geometry
{\displaystyle \mathbb {G} (1,3)} . In order to get the Euler class, the total Chern class of T ∗ {\displaystyle T^{*}} must be computed, which is given
Schubert_calculus
Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern
Segre_class
Generalization of vector bundles
+c_{i-1}(A)c_{1}(C)+c_{i}(C).} It follows that the Chern classes of a vector bundle E {\displaystyle E} depend only on the class of E {\displaystyle E} in the Grothendieck
Coherent_sheaf
Topological quantum field theory
after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional
Chern–Simons_theory
Short exact sequence of sheaves on projective space
{E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0,} we can compute the total Chern class of E {\displaystyle {\mathcal {E}}} with the formula c ( E ) = c ( E
Euler_sequence
Unsolved problem in geometry
Hodge classes than the Chern classes of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes. Consequently
Hodge_conjecture
Characteristic class of oriented, real vector bundles
Thom isomorphism Generalized Gauss–Bonnet theorem Chern class Pontryagin class Stiefel–Whitney class Milnor & Stasheff 74, Property 9.2 Milnor & Stasheff
Euler_class
Mathematical concept
up to isomorphism by their Chern classes, which are integers: they lie in H2(CPn,Z) = Z. In fact, the first Chern classes of complex projective space
Complex_projective_space
Kind of complex manifold
\mathbb {Z} )} is the first Chern class map, sending an isomorphism class of a line bundle to its associated first Chern class. It turns out that there is
Complex_torus
Set of topological invariants
_{t=0}^{i}{j+t-i-1 \choose t}w_{i-t}w_{j+t}.} Characteristic class for a general survey, in particular Chern class, the direct analogue for complex vector bundles
Stiefel–Whitney_class
forms are important because of the following: The alternatization of the Chern class of any factor of automorphy is a Riemann form. Conversely, given any
Riemann_form
Combination of higher category theory with Chern–Weil theory
In mathematics, ∞-Chern–Weil theory is a generalized formulation of Chern–Weil theory from differential geometry using the formalism of higher category
∞-Chern–Weil_theory
gth Chern class of the Hodge bundle, appearing in its integrand. The other factor is a monomial in the ψ i {\displaystyle \psi _{i}} , the first Chern classes
Lambda_g_conjecture
Algebraic topology theory
first Chern class; hence, it belongs to the completion of the equivariant cohomology ring.) In the non-equivariant case, the first Chern class can be
Equivariant_cohomology
Concept in differential geometry
manifold X {\displaystyle X} the second Stiefel-Whitney class can be computed as the first Chern class mod 2 {\displaystyle {\text{mod }}2} . A genus g Riemann
Spin_structure
Type of metric in Riemannian geometry
three cases dependent on the sign of the first Chern class of the Kähler manifold: When the first Chern class is negative, there is always a Kähler–Einstein
Kähler–Einstein_metric
Riemannian metrics, complex manifolds
According to Chern–Weil theory, the Ricci form of any such metric is a closed differential 2-form which represents the first Chern class. Calabi conjectured
Calabi_conjecture
Chinese-American mathematician (born 1949)
first Chern class. A proposal of Calabi's suggested that Kähler–Einstein metrics exist on any compact Kähler manifolds with positive first Chern class which
Shing-Tung_Yau
Vector bundle of rank 1
smooth structures (and thus the same first Chern class) but different holomorphic structures. The Chern class statements are easily proven using the exponential
Line_bundle
Analogs of homology groups for algebraic varieties
scheme X over a field has Chern classes ci(E) in CHi(X), with the same formal properties as in topology. The Chern classes give a close connection between
Chow_group
Type of surface in algebraic geometry
surface intersection of {w=0} and F, therefore we recover that the second Chern class of S equals 27. b) Let w1, w2 be two 1-forms on S. The canonical divisor
Fano_surface
complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle
Complex_vector_bundle
all Kähler manifolds. Let X be a compact Kähler manifold. The first Chern class c1 gives a map from holomorphic line bundles to H2(X, Z). By Hodge theory
Lefschetz theorem on (1,1)-classes
Lefschetz_theorem_on_(1,1)-classes
Secondary characteristic classes of 3-manifolds
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons
Chern–Simons_form
Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class c 1 ( L ) = [ ω ] {\displaystyle c_{1}(L)=[\omega ]} nef and big has
Fujiki_class_C
for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern classes. Giambelli's formula
Porteous_formula
Branch of mathematics
+x_{n}^{m}).} The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. The Chern character is used
K-theory
French mathematician (1928–2014)
stack Approximation property – Mathematical concept Barsotti–Tate group Chern class Crystal (mathematics) Crystalline cohomology – Weil cohomology theory
Alexander_Grothendieck
Topics referred to by the same term
Chern may refer to: Shiing-Shen Chern (1911–2004), Chinese-American mathematician Chern class, a type of characteristics class associated to complex vector
Chern_(disambiguation)
Generalizes the Kodaira vanishing theorem
(p,0)-forms taking values on F. The theorem states that, if the first Chern class of F is negative, H q ( M ; Ω p ( F ) ) = 0 when q + p < n . {\displaystyle
Nakano_vanishing_theorem
Chinese mathematician (born 1958)
had settled the case of closed Kähler manifolds with nonpositive first Chern class. His work in applying the method of continuity showed that C0 control
Tian_Gang
bundles on M. The connecting homomorphism sends a line bundle to its first Chern class. Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry
Exponential_sheaf_sequence
Study of vector bundles, principal bundles, and fibre bundles
unitary group SU ( 2 ) {\displaystyle \operatorname {SU} (2)} and second Chern class c 2 ( P ) = 1 {\displaystyle c_{2}(P)=1} , then the moduli space M P
Gauge_theory_(mathematics)
Tensor in differential geometry
Ricci form is a closed 2-form. Its cohomology class is, up to a real constant factor, the first Chern class of the canonical bundle, and is therefore a
Ricci_curvature
Vector bundle existing over a Grassmannian
generator of negative degree. Hopf bundle Stiefel-Whitney class Euler sequence Chern class (Chern classes of tautological bundles is the algebraically independent
Tautological_bundle
Theory in theoretical physics
exists when the first Chern classes of associated bundles sum to zero whereas the A model exists when the difference of the Chern classes is zero. In the Kähler
Topological_string_theory
On when a definite intersection form of a smooth 4-manifold is diagonalizable
{M}}=8k-3(1-b_{1}(X)+b_{+}(X)),} where k = c 2 ( P ) {\displaystyle k=c_{2}(P)} is a Chern class, b 1 ( X ) {\displaystyle b_{1}(X)} is the first Betti number of X {\displaystyle
Donaldson's_theorem
this class. Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers
Surface_of_general_type
Partial differential equations whose solutions are instantons
connection), then the underlying principal bundle must have trivial Chern classes, which is a topological obstruction to the existence of flat connections:
Yang–Mills_equations
Result in algebraic geometry
bundles. Using this isomorphism, consider the Chern character (a rational combination of Chern classes) as a functorial transformation: c h : K 0 ( X
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Fiber bundle whose fibers are projective spaces
+c_{r}(E)=0} where ci(E) is the i-th Chern class of E. One interesting feature of this description is that one can define Chern classes as the coefficients in the
Projective_bundle
Type of smooth complex surface of kodaira dimension 0
{\displaystyle c_{i}(X)} is the i-th Chern class of the tangent bundle. Since K X {\displaystyle K_{X}} is trivial, its first Chern class c 1 ( K X ) = − c 1 ( X )
K3_surface
Manifold with Riemannian, complex and symplectic structure
first Chern class of L {\displaystyle L} in H 2 ( X , Z ) {\displaystyle H^{2}(X,\mathbb {Z} )} . Any Kähler form ω {\displaystyle \omega } whose class in
Kähler_manifold
Mathematical technique for vector bundles
bundles, one often wishes to simplify computations, for example of Chern classes. Often computations are well understood for line bundles and for direct
Splitting_principle
Metric on a determinant line bundle
determinant line bundle. It can be seen as defining the Chern–Weil representative of the first Chern class of this ample line bundle. The Quillen metric construction
Quillen_metric
Mathematical theory
Chow groups. The arithmetic Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic
Arakelov_theory
Surname list
Minister of Health Chern Shiing-Shen (陳省身; 1911–2004), Chinese-American mathematician, known for Chern–Gauss–Bonnet theorem, Chern class, Chern–Simons theory
Chen_(surname)
3d hypersurface of degree 5
descends to a vector bundle on this projective Grassmannian. Its total Chern class is c ( T ∗ ) = 1 + σ 1 + σ 1 , 1 {\displaystyle c(T^{*})=1+\sigma _{1}+\sigma
Quintic_threefold
Mathematical physics relation
geometry and before Chern had made history with his contributions to the generalized Gauss–Bonnet theorem and the Chern classes.) We had much to talk
Wu–Yang_dictionary
Special type of principal bundle
) A corresponding isomorphism is given by the first Chern class. Although characteristic classes are defined for vector bundles, it is possible to also
Principal_U(1)-bundle
French mathematician (1942–2009)
with Yau, he also showed that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the
Thierry_Aubin
Principal fiber bundle
H^{2}(M)} . This isomorphism is realized by the Euler class; equivalently, it is the first Chern class of a smooth complex line bundle (essentially because
Circle_bundle
Breakdown of parity at the quantum level
answer h times the second Chern class of the gauge bundle over M × S 1 {\displaystyle M\times S^{1}} . This second Chern class may be any integer. In particular
Parity_anomaly
Subject area in mathematics
K-theory and then apply the Chern character and Todd class of Y, or one can first apply the Chern character and Todd class of X and then compute the pushforward
Algebraic_K-theory
Smooth manifold
geometryPages displaying short descriptions of redirect targets Chern class – Characteristic classes of vector bundles Frölicher–Nijenhuis bracket Kähler manifold –
Almost_complex_manifold
{Spin} ^{\mathrm {c} }(M)\rightarrow \mathbb {Z} } (hence the first Chern class c 1 : Spin c ( M ) → H 2 ( M , Z ) {\displaystyle c_{1}\colon \operatorname
Kronheimer–Mrowka_basic_class
the Hodge vector bundle and c(E*) the total Chern class of its dual vector bundle; ψi is the first Chern class of the cotangent line bundle to the i-th marked
ELSV_formula
{\displaystyle \mathbb {P} (V)} ), there is the natural identification (see Chern class#Complex projective space for example): Hom ( l , V / l ) = T l P (
Grassmann_bundle
Superconductivity theory
{\displaystyle c_{1}(L)=c_{1}(L)[\Sigma ]\in H^{2}(\Sigma )} is the first Chern class. The Lagrangian is minimized (stationary) when ψ , A {\displaystyle \psi
Ginzburg–Landau_theory
Concept in algebraic geometry
j=1,2} cases of this vanishing statement also tell us that the first Chern class induces an isomorphism c 1 : P i c ( X ) → H 2 ( X , Z ) {\displaystyle
Fano_variety
Lie group of complex numbers of unit modulus; topologically a circle
suitable spaces, the isomorphism classes of principal U ( 1 ) {\displaystyle U(1)} -bundles are classified by the first Chern class, c 1 ∈ H 2 ( X ; Z ) {\displaystyle
Circle_group
Continuous surjection satisfying a local triviality condition
sphere bundle is called a circle bundle and the Euler class is equal to the first Chern class, which characterizes the topology of the bundle completely
Fiber_bundle
Manifold
a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes. Complex dimension Complex analytic variety Quaternionic manifold
Complex_manifold
Moduli space of the Seiberg–Witten equations
L=\det(W^{\pm })} . Since the determinant line bundle preserves the first Chern class, one has c 1 ( s ) := c 1 ( L ) = c 1 ( W ± ) {\displaystyle c_{1}({\mathfrak
Seiberg–Witten_moduli_space
Solitons in Euclidean spacetime
characteristic class. If the gauge symmetry is a unitary group or special unitary group then this characteristic class is the second Chern class, which vanishes
Instanton
1974 textbook by John Milnor and James Stasheff
basic characteristic classes is described, which includes Stiefel–Whitney classes, Chern classes, Pontrjagin classes and the Euler class. Afterwards, it descibres
Characteristic_Classes
the fundamental class [ X ] ∈ H 4 ( X , Z ) {\displaystyle [X]\in H_{4}(X,\mathbb {Z} )} from the orientation, the second Chern class concretely gives
Donaldson_invariant
Russian-French mathematician
G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101–172. Grisha Perelman. The entropy
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Make a meromorphic function from local data in multiple variables
the additive problem, meets an obstruction in the form of the first Chern class (see also exponential sheaf sequence). In terms of sheaf theory, let
Cousin_problems
Concept in algebraic geometry
other direction, for a line bundle L on a projective variety, the first Chern class c 1 ( L ) {\displaystyle c_{1}(L)} means the associated Cartier divisor
Ample_line_bundle
with the first Chern class of X {\displaystyle X} is negative. Such configurations make the moduli space very singular so a fundamental class cannot be defined
Kuranishi_structure
Mathematical property of a space
bundlesPages displaying short descriptions of redirect targets Chern class – Characteristic classes of vector bundles Euler characteristic – Topological invariant
Topological_property
Introduction and Reference on Differential Geometry
of characteristic classes of principal bundles (Chern–Weil theory), it covers Euler classes, Chern classes, and Pontryagin classes. The second volume
Foundations of Differential Geometry
Foundations_of_Differential_Geometry
Vector bundles theorem
[\omega ]^{n-1})[X]} where c 1 ( E ) {\displaystyle c_{1}(E)} is the first Chern class of E {\displaystyle E} . The slope of E {\displaystyle E} is the rational
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
obtaining the solutions to the equations; The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e., c 2 ( M
Strominger's_equations
Mathematical space
integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of E {\displaystyle E} . In particular, all of the integral cohomology
Grassmannian
moduli of vector bundles of rank r = 2 {\displaystyle r=2} and first Chern class c 1 = 0 {\displaystyle c_{1}=0} on the complex projective line P 1 {\displaystyle
Stable_vector_bundle
Type of Kac–Moody algebras
over this free loop group, which are classified by a two-class known as the first Chern class of the fibration. Therefore, the central extensions of an
Affine_Lie_algebra
account for its curvature. Chern class: Chern classes are characteristic classes in mathematics first introduced by Shiing-Shen Chern in 1946. Chow's moving
List_of_Chinese_discoveries
Concept in algebraic geometry
every curve C in X. To go back from line bundles to divisors, the first Chern class is the isomorphism from the Picard group of line bundles on a variety
Nef_line_bundle
Mathematical theorem
invertible sheaves (line bundles) the second Chern class vanishes. The products of second cohomology classes can be identified with intersection numbers
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Phase of a cycle
\alpha =0} of electrons in graphene. Berry connection and curvature Chern class Maslov index Optical rotation Quantum geometry in condensed-matter physics
Geometric_phase
a Hermitian Yang–Mills connection (or Hermite–Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle
Hermitian Yang–Mills connection
Hermitian_Yang–Mills_connection
Diffeomorphism Large diffeomorphism Orientability characteristic class Chern class Pontrjagin class spin structure differentiable map submersion immersion Embedding
List of differential geometry topics
List_of_differential_geometry_topics
Mathematical inequality
general type, and let c1 = c1(X) and c2 = c2(X) be the first and second Chern class of the complex tangent bundle of the surface. Then c 1 2 ≤ 3 c 2 . {\displaystyle
Bogomolov–Miyaoka–Yau inequality
Bogomolov–Miyaoka–Yau_inequality
Conjecture in algebraic geometry
dimMg,n = 3g – 3 + n, and 0 if no such g exists, where c1 is the first Chern class of a line bundle. Witten's generating function F ( t 0 , t 1 , … ) =
Witten_conjecture
Concept in algebraic geometry
2\int _{A}c_{1}(TX)} , where c 1 {\displaystyle c_{1}} is the first Chern class of the tangent bundle TX, regarded as a complex vector bundle by choosing
Quantum_cohomology
of the d-th Newton polynomial sd, evaluated on the (algebraic) total Chern class of the tangent bundle of V. This number s d ( V ) {\displaystyle s_{d}(V)\
Norm_variety
Special type of principal bundle
the second Chern class. If B {\displaystyle B} is again a 4-manifold, then the classification is unique. Although characteristic classes are defined
Principal_SU(2)-bundle
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