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Construct in algebraic geometry
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric
Cotangent_complex
Mathematical object
{\text{Ext}}^{1}(\mathbf {L} _{X_{0}/k},{\mathcal {O}}_{X_{0}})} . Cotangent complex Schlessinger's theorem characteristic linear system of an algebraic
Kodaira–Spencer_map
homotopy equivalence in a suitable model category. The (relative) cotangent complex of an ( A ∙ , d ) {\displaystyle (A_{\bullet },d)} -differential graded
Derived_scheme
Vector bundle of cotangent spaces at every point in a manifold
especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold
Cotangent_bundle
In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules
Cotangent_sheaf
from the ring morphism. Deformation theory Cotangent complex Picard stack Illusie, Luc. Complexe Cotangent et Deformations I. pp. 151–168. Tangent Spaces
Exalcomm
French mathematician
work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012
Luc_Illusie
Functions of an angle
Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less commonly used. Each of these six trigonometric
Trigonometric_functions
Theory of cohomology for commutative rings
cohomology for commutative rings which is closely related to the cotangent complex. The first three cohomology groups were introduced by Stephen Lichtenbaum
André–Quillen_cohomology
Hyperbolic analogues of trigonometric functions
derived: hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/), hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/), hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/), hyperbolic
Hyperbolic_functions
Fundamental trigonometric functions
the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal
Sine_and_cosine
Differential form in commutative algebra
cotangent sheaf can be computed from the sheafification of the cotangent module on the underlying graded algebra. For example, consider the complex curve
Kähler_differential
Inverse functions of sin, cos, tan, etc.
domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from
Inverse trigonometric functions
Inverse_trigonometric_functions
Theory for associative algebras over rings
using a self-intersection from the diagonal, or more generally, the cotangent complex L X / S ∙ {\displaystyle \mathbf {L} _{X/S}^{\bullet }} since this
Hochschild_homology
{\textbf {L}}_{X}} , where L X {\displaystyle {\textbf {L}}_{X}} is the cotangent complex of X, that induces an isomorphism on h 0 {\displaystyle h^{0}} and
Perfect_obstruction_theory
French mathematician (1928–2014)
Hakim [fr] (relative schemes and classifying topos), Luc Illusie (cotangent complex), Michel Raynaud, Michèle Raynaud, Jean-Louis Verdier (co-founder
Alexander_Grothendieck
Smooth manifold
complexified tangent and cotangent bundles. In both cases one demands that the direct sum of the subbundle and its complex conjugate yield the original
Almost_complex_manifold
_{R}=\Omega _{Q(R)}^{1}\otimes _{Q(R)}^{L}R} is an R-module called the cotangent complex of R. It is functorial in R: each R → S gives rise to L R → L S {\displaystyle
Derived_tensor_product
Branch of mathematics
(or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications
Derived_algebraic_geometry
Branch of mathematics
Exalcomm Cotangent complex Gromov–Witten invariant Moduli of algebraic curves Degeneration (algebraic geometry) Palamodov (1990). "Deformations of Complex Spaces"
Deformation_(mathematics)
Circle with radius of one
fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant –
Unit_circle
Type of manifold in differential geometry
abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical
Symplectic_manifold
French mathematician (1932–2024)
Discrete Groups and Renormalization. Springer. ISBN 9783540303084. Cotangent complex Dieudonné module MacMahon's master theorem "Pierre Cartier". Institute
Pierre Cartier (mathematician)
Pierre_Cartier_(mathematician)
Manifold upon which it is possible to perform calculus
The cotangent space at a point is the dual of the tangent space at that point and the elements are referred to as cotangent vectors; the cotangent bundle
Differentiable_manifold
Pairing in algebra between ext groups of modules
and L X / Y {\displaystyle \mathbf {L} _{X/Y}} corresponds to the cotangent complex. Ext functor Derived category Deformation theory Kodaira–Spencer map
Yoneda_product
Concept in category theory
in groupoids. One notable example of this is in the study of the cotangent complex for local-complete intersections and in the study of exalcomm. Grothendieck
Fibred_category
Concept in string theory
genus g is the generating function of the genus g GW invariants. Cotangent complex – for deformation theory Schubert calculus Hori, Kentaro (2003). Mirror
Gromov–Witten_invariant
Mathematical conjecture
Hodge structures. Further, these integrals are actually computable. Cotangent complex Homotopy associative algebra Kuranishi structure Mirror symmetry (string
Mirror_symmetry_conjecture
Scheme in algebraic geometry
{\displaystyle k} . If L X {\displaystyle {\textbf {L}}_{X}} denotes the cotangent complex of X relative to k {\displaystyle k} , then the intrinsic normal bundle
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
complex-valued one-forms d z j {\displaystyle dz^{j}} and d z ¯ j {\displaystyle d{\bar {z}}^{j}} provide the splitting of the complexified cotangent
Holomorphic_tangent_bundle
Principal square root of minus 1
Euler expressed the partial fraction decomposition of the trigonometric cotangent as π cot π z = 1 z + 1 z − 1 + 1 z + 1 + 1 z − 2 + 1 z + 2 + ⋯ . {\textstyle
Imaginary_unit
Algebraic structure in linear algebra
O. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space.
Vector_space
mathematics Prove or disprove: Any closed exact Lagrangian submanifold of the cotangent bundle of a closed manifold is Hamiltonian isotopic to the zero section
Nearby_Lagrangian_conjecture
Mathematical functions
hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions
Inverse_hyperbolic_functions
example is the cotangent complex for a local complete intersection scheme X {\displaystyle X} which is given by the two-term complex L X ∙ = i ∗ I /
Abelian_2-group
the cotangent space to the Riemann moduli space, or Teichmüller space. Each quadratic differential on a domain U {\displaystyle U} in the complex plane
Quadratic_differential
Elliptic differential operators in geometry mathematics
(Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian-
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Y} is a any local complete intersection morphism of schemes, its cotangent complex L X / Y {\displaystyle L_{X/Y}} is perfect of Tor-amplitude [-1,0]
Regular_embedding
Mathematical functions related to Weierstrass's elliptic function
to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative
Weierstrass sigma, zeta, and eta functions
Weierstrass_sigma,_zeta,_and_eta_functions
Area of geometry, about angles and lengths
reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively: csc A = 1 sin A = hypotenuse opposite = h a
Trigonometry
Property of a differential manifold that includes complex structures
vector field on M. The cotangent bundle of M, denoted T*, is the vector bundle over M whose sections are one-forms on M. In complex geometry one considers
Generalized_complex_structure
microfunction microlocal The notion microlocal refers to a consideration on the cotangent bundle to a space as opposed to that on the space itself. Explicitly,
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Assignment of vector fields to manifolds
I / I 2 {\displaystyle I/I^{2}} can be shown to be isomorphic to the cotangent space T x ∗ M {\displaystyle T_{x}^{*}M} through the use of Taylor's theorem
Tangent_space
Type of Riemannian manifold
showed the more general statement that cotangent bundle T ∗ C P n {\displaystyle T^{*}\mathbb {CP} ^{n}} of any complex projective space has a complete hyperkähler
Hyperkähler_manifold
Riemannian manifold with SU(n) holonomy
the fibers of the vector bundle. Using this, we can use the relative cotangent sequence 0 → p ∗ Ω C → Ω V → Ω V / C → 0 {\displaystyle 0\to p^{*}\Omega
Calabi–Yau_manifold
Branch of mathematics
full Bloch–Kato conjecture. Derived algebraic geometry Derivator Cotangent complex - one of the first objects discovered using homotopical algebra L∞
Homotopical_algebra
Differentiable manifold
characterized in terms of duality. Consider the line subbundle of the complex cotangent bundle annihilating V H 0 M = V ∗ = ( L ⊕ L ¯ ) ⊥ ⊂ T ∗ M ⊗ C . {\displaystyle
CR_manifold
This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will
List of trigonometric identities
List_of_trigonometric_identities
graded of the conjugate filtration) and the exterior powers of the cotangent complex. Pierre Deligne; Luc Illusie (1987). "Relèvements modulo p2 et décomposition
Cartier_isomorphism
Topos-theoretic version of a ringed space
has applications to deformation theory in algebraic geometry (cf. cotangent complex) and the mathematical foundation of quantum mechanics. In the latter
Ringed_topos
Mathematical concept
an n-manifold, considered as a 2n-manifold, has an almost complex structure, and the cotangent bundle of an n-manifold, considered as a 2n-manifold, has
Symplectic_vector_space
Number which when multiplied by x equals 1
The trigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the
Multiplicative_inverse
Complex vector bundle on a complex manifold
examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one
Holomorphic_vector_bundle
Theorem in algebraic geometry
K_{X}} to be the bundle of n-forms on X, the top exterior power of the cotangent bundle: K X = Ω X n = ⋀ n ( T ∗ X ) . {\displaystyle K_{X}=\Omega _{X}^{n}={\bigwedge
Serre_duality
Equation for radii of tangent circles
geodesic curvature of the circle relative to the sphere, which equals the cotangent of the oriented intrinsic radius ρ j . {\displaystyle \rho _{j}.} Then:
Descartes's_theorem
contravariant functor with values in Abelian groups. André–Quillen cohomology Cotangent complex H-space Quillen, Dan. "On the (co-) homology of commutative rings"
H-object
Technique in integral evaluation
\left|\cos x\right|+C\\&=\ln \left|\sec x\right|+C.\end{aligned}}} The cotangent function can be integrated similarly by expressing it as cot x = cos
Integration_by_substitution
Concept in algebraic geometry
{\displaystyle n} th exterior power of the cotangent bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over the complex numbers, it is the determinant bundle
Canonical_bundle
Module over a sheaf of differential operators
the symbols, which in the good case is a Lagrangian submanifold of the cotangent bundle of maximal dimension (involutive systems). The techniques were
D-module
Here π is the projection of the cotangent bundle T*M to M, and π* is the pullback of a vector bundle. Chain complex Atiyah, M. F.; Singer, I. M. (1968)
Elliptic_complex
Mathematical manifold theory
extending (see Gramian matrix) the inner product induced by g from each cotangent fiber T p ∗ ( M ) {\displaystyle T_{p}^{*}(M)} to its k t h {\displaystyle
Hodge_theory
Class of mathematical expression
It is the natural way to view the range of the tangent function and cotangent functions of trigonometry: tan x {\displaystyle \tan x} approaches
Division_by_zero
Space of possible positions for all objects in a physical system
the velocities of the points q ∈ Q {\displaystyle q\in Q} , while the cotangent space T ∗ Q {\displaystyle T^{*}Q} corresponds to momenta. (Velocities
Configuration_space_(physics)
for all (complex) even dimensional smooth complete intersections in C P n {\displaystyle \mathbb {CP} ^{n}} . For example, consider the cotangent bundle
Atiyah–Hirzebruch spectral sequence
Atiyah–Hirzebruch_spectral_sequence
Canonical differential form
mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of a manifold Q . {\displaystyle Q
Tautological_one-form
Vector bundle of rank 1
{\displaystyle V} . This construction is in particular applied to the cotangent bundle of a smooth manifold. The resulting determinant bundle (more precisely
Line_bundle
Triangle area in terms of side lengths
}{2}}\cot {\tfrac {\beta }{2}}\cot {\tfrac {\gamma }{2}}} , the triple cotangent identity, which applies because the sum of half-angles is α 2 + β 2 +
Heron's_formula
\beta \rangle } is the inner product on the exterior products of the cotangent space of X {\displaystyle X} induced by the Riemannian metric. Using this
Kähler_identities
Type of surface in algebraic geometry
powerful applications. By example, one can recover the fact that the cotangent space of S is generated by global sections. This space of global 1-forms
Fano_surface
Structure defining distance on a manifold
that g⊗ is regarded also as a section of the bundle T*M ⊗ T*M of the cotangent bundle T*M with itself. Since g is symmetric as a bilinear mapping, it
Metric_tensor
Characteristic classes of vector bundles
0}dz_{i}-z_{i}dz_{0} \over z_{0}^{2}},\,i\geq 1.} In other words, the cotangent sheaf Ω C P n | U {\displaystyle \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}}
Chern_class
Branch of geometry
elements of M {\displaystyle M} can be identified with a quotient of the cotangent bundle T ∗ M {\displaystyle T^{*}M} (with the zero section 0 M {\displaystyle
Contact_geometry
Parametrizes complex structures on a surface
differentials on a Riemann surface X {\displaystyle X} are identified with the cotangent space at ( X , f ) {\displaystyle (X,f)} to Teichmüller space. The Weil–Petersson
Teichmüller_space
manifold Tensor analysis Tangent vector Tangent space Tangent bundle Cotangent space Cotangent bundle Tensor Tensor bundle Vector field Tensor field Differential
List of differential geometry topics
List_of_differential_geometry_topics
Mathematical result in differential geometry
coordinate charts, and is a function on the cotangent bundle of X, homogeneous of degree n on each cotangent space. (In general, differential operators
Atiyah–Singer_index_theorem
inverse cosecant function. (Also written as arccsc.) arccot – inverse cotangent function. arccsc – inverse cosecant function. (Also written as arccosec
List of mathematical abbreviations
List_of_mathematical_abbreviations
Typically linear operator defined in terms of differentiation of functions
principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle). More generally, let E and F be vector bundles over a manifold
Differential_operator
Space of all possible states that a system can take
phase space. More abstractly, in classical mechanics phase space is the cotangent bundle of configuration space, and in this interpretation the procedure
Phase_space
surface F to its Albanese variety A which induces a homomorphism from the cotangent space of the Albanese variety (of dimension q) to H1,0(F). Jun-Ichi Igusa
Irregularity_of_a_surface
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
operating on complex numbers represented by their polar coordinates, especially if the magnitude of the numbers is not relevant (multiplying a complex vector
CORDIC
vanishing at x, then the cotangent space at x is mx / mx2. The tangent space is (mx / mx2)*, the dual vector space to the cotangent space. Analytic mappings
Analytic_space
Mathematical concept
varieties and their cotangent bundles. In the 21st century, this approach evolved into a more general framework where the traditional cotangent bundle of the
Symplectic_resolution
Connected sum Connection Cotangent bundle – the vector bundle of cotangent spaces on a manifold. Cotangent space Covering Cusp CW-complex Dehn twist Diffeomorphism
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
Italian-born American mathematician (1923–2023)
Kähler–Einstein metric on the total space. In the case of the cotangent bundle of a complex space form, one obtains a hyperkähler metric. The Eguchi–Hanson
Eugenio_Calabi
Mathematical notation based on the Arabic script
is used in some regions (e.g. Syria); Arabic for "tangent" is ظل ẓill Cotangent cot {\displaystyle \cot } طتا from طتا ṭāʾ (i.e. dotless ظ ẓāʾ)-tāʾ-ʾalif;
Modern Arabic mathematical notation
Modern_Arabic_mathematical_notation
Concept in algebraic geometry
{I}}^{2}\to i^{*}\Omega _{X}\to \Omega _{Y}\to 0,} where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism
Adjunction_formula
Expression that may be integrated over a region
a smooth section of the k {\displaystyle k} th exterior power of the cotangent bundle of M {\displaystyle M} . The set of all differential k {\displaystyle
Differential_form
Arrangement of information or data, typically in rows and columns
information or data, typically in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data
Table_(format)
Field of algebraic geometry
line bundle of n-forms KX = Ωn, which is the nth exterior power of the cotangent bundle of X. For an integer d, the dth tensor power of KX is again a line
Birational_geometry
It is a generalization of the Lie bracket from an operation on the tangent bundle
transformation is known in the physics literature as a shift in the B field. The cotangent bundle, T ∗ {\displaystyle {\mathbf {T} }^{*}} of M {\displaystyle M}
Courant_bracket
Theory proposed by Roger Penrose
rays or massless particles and can be regarded as a complexification or cotangent bundle of the original twistor description. By extending the ambitwistor
Twistor_theory
stably free module is free of unique rank Hermite-Sobolev spaces Hermite's cotangent identity, a trigonometric identity Hermite's criterion Hermite's identity
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Assignment of a tensor continuously varying across a region of space
a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a
Tensor_field
Sheaf of rings in mathematics
{m}}_{x}/{\mathfrak {m}}_{x}^{2}} is a vector space over that field (the cotangent space). The tangent space T x ( X ) {\displaystyle T_{x}(X)} is defined
Ringed_space
Extension of superfactorials to the complex numbers
evaluation of the cotangent integral: introducing the notation Lc ( z ) {\displaystyle \operatorname {Lc} (z)} for the log-cotangent integral, and using
Barnes_G-function
Critical point where a periodic solution arises
given by a Lie derivative defined on the tangent bundle. Because all cotangent bundles are always symplectic manifolds, it is common to formulate bifurcation
Hopf_bifurcation
Correspondsnce between Higgs bundles and fundamental group representations
structure on the cotangent bundle to the Jacobian. It is possible to define the notion of a principal G {\displaystyle G} -Higgs bundle for a complex reductive
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
counter-example to the statement: Any closed exact Lagrangian submanifold of the cotangent bundle of a closed manifold is Hamiltonian isotopic to the zero section
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematical formula
mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite. Suppose a1, ..., an are complex numbers, no two of which
Hermite's_cotangent_identity
Mathematical functions
hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1}
Lemniscate_elliptic_functions
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