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Ruled surface over the projective line
mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by Friedrich Hirzebruch (1951). The Hirzebruch surface Σ n {\displaystyle
Hirzebruch_surface
In mathematics, a Inoue–Hirzebruch surface is a complex surface with no meromorphic functions introduced by Inoue (1977). They have Kodaira dimension
Inoue–Hirzebruch_surface
German mathematician (1927–2012)
Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds
Friedrich_Hirzebruch
On the Euler characteristic of a holomorphic vector bundle on a compact complex manifold
theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch_theorem
Mathematical classification of surfaces
the Hirzebruch surfaces Σn for n = 0 or n ≥ 2. (The Hirzebruch surface Σn is the P1 bundle over P1 associated to the sheaf O(0) + O(n). The surface Σ0
Enriques–Kodaira classification
Enriques–Kodaira_classification
Analogs of homology groups for algebraic varieties
^{r-1}+c_{2}\zeta ^{r-2}+\cdots +c_{r}}}} For example, the Chow ring of a Hirzebruch surface can be readily computed using the projective bundle formula. Recall
Chow_group
Surface in algebraic geometry
for Hirzebruch surfaces and greater than 1 for other rational surfaces. The Picard group is the odd unimodular lattice I1,n, except for the Hirzebruch surfaces
Rational_surface
Concept in algebraic geometry
such surface, given by blowing up the projective plane in 2 distinct points. Degree 8: they have 2 isomorphism types. One is a Hirzebruch surface given
Del_Pezzo_surface
Algebraic surface in mathematics
papers Hirzebruch (1971), Hirzebruch & Van de Ven (1974) and Hirzebruch & Zagier (1977) identified their type in the classification of algebraic surfaces. Most
Hilbert_modular_variety
surfaces Del Pezzo surfaces, surfaces with an ample anticanonical divisor Hirzebruch surfaces, rational ruled surfaces Segre surfaces, intersections of
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
Mathematical theorem
due to Hirzebruch. One form of the Riemann–Roch theorem states that if D {\displaystyle D} is a divisor on a non-singular projective surface then χ (
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Typeface style used in mathematics
ISBN 978-1-58115-320-0. Gunning, Robert C. (1966). Lectures on Riemann Surfaces. Mathematical Notes. Princeton University Press. p. 1. Narasimhan, Raghavan
Blackboard_bold
Surface containing a line through every point
bundle over some curve. The ruled surfaces with base curve of genus 0 are the Hirzebruch surfaces. Doubly ruled surfaces are the inspiration for curved hyperboloid
Ruled_surface
Algebro-geometric stability condition
\mathbb {F} _{1}=\operatorname {Bl} _{0}\mathbb {CP} ^{2}} , the first Hirzebruch surface, which is the blow up of the complex projective plane at a point,
K-stability
Complex projective plane Del Pezzo surface E8 manifold Enriques surface Exotic R4 Hirzebruch surface K3 surface For more examples see 4-manifold. Brieskorn
List_of_manifolds
Part of the Kodaira classification
Inoue–Hirzebruch surfaces, Enoki surfaces, and Kato surfaces give examples of type VII surfaces with b2 > 0. The minimal class VII surfaces with second
Surface_of_class_VII
of Kato surfaces include Inoue-Hirzebruch surfaces and Enoki surfaces. The global spherical shell conjecture claims that all class VII surfaces with positive
Kato_surface
projective planes. Barthel, Hirzebruch & Höfer (1987) gave a method for finding examples, which in particular produced a surface X with c2 1 = 3c2 = 3254
Bogomolov–Miyaoka–Yau inequality
Bogomolov–Miyaoka–Yau_inequality
Study of complex manifolds and several complex variables
techniques arising out of differential geometry and analysis. For example, the Hirzebruch-Riemann-Roch theorem, a special case of the Atiyah-Singer index theorem
Complex_geometry
Theorem
Then the theorem states that S is rational. Hirzebruch surface List of complex and algebraic surfaces Castelnuovo’s Theorem Kurke, G. (1972). "The castelnuovo
Noether's theorem on rationality for surfaces
Noether's_theorem_on_rationality_for_surfaces
Non-singular cubic surface in mathematics
unity. The surface has 10 Eckardt points where 3 lines meet, given by the point (1: −1: 0: 0: 0) and its conjugates under permutations. Hirzebruch (1976)
Clebsch_surface
Relation between genus, degree, and dimension of function spaces over surfaces
An n-dimensional generalisation, the Hirzebruch–Riemann–Roch theorem, was found and proved by Friedrich Hirzebruch, as an application of characteristic
Riemann–Roch_theorem
Algebraic variety containing an algebraic torus
blow-ups starting from either the complex projective plane, or from a Hirzebruch surface. Every toric variety has a resolution of singularities given by another
Toric_variety
Branch of mathematics
This make it possible to compute the K 0 {\displaystyle K_{0}} or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group
K-theory
Result in algebraic geometry
far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Fiber bundle whose fibers are projective spaces
cone (algebraic geometry) ruled surface (an example of a projective bundle) Severi–Brauer variety Hirzebruch surface Hartshorne 1977, Ch. II, Exercise
Projective_bundle
Number of "holes" of a surface
568–576. doi:10.1016/0196-6774(89)90006-0. ISSN 0196-6774. Zbl 0689.68071. Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics
Genus_(mathematics)
Ring homomorphism from the cobordism ring of manifolds to another ring
"Computing Hirzebruch L-Polynomials". Huybrechts, Daniel. "14.1 Existence, uniqueness, and embeddings of lattices". Lectures on K3 Surfaces (PDF). p. 285
Genus of a multiplicative sequence
Genus_of_a_multiplicative_sequence
Classification of symplectic toric manifolds
segments, i.e. a rectangle in R 2 {\displaystyle \mathbb {R} ^{2}} . Any Hirzebruch surface is a 4-dimensional toric manifold; its associated Delzant polytope
Delzant's_theorem
British-Lebanese mathematician (1929–2019)
Donnelly and I. Singer, he extended Hirzebruch's formula (relating the signature defect at cusps of Hilbert modular surfaces to values of L-functions) from
Michael_Atiyah
the signature theorem. Hirzebruch (1973) introduced the signature defect for the cusp singularities of Hilbert modular surfaces. Michael Francis Atiyah
Signature_defect
NonKaehler Geometry, Sapporo, 2008 March. G. Dloussky, "Une construction elementaire des surfaces d'Inoue–Hirzebruch". Math. Ann. 280, 663–682 (1988).
Inoue_surface
American mathematician
with Hirzebruch in work on Hilbert modular surfaces. Hirzebruch and Zagier coauthored Intersection numbers of curves on Hilbert modular surfaces and modular
Don_Zagier
Japanese mathematician (1915–1997)
available. This work was particularly influential, for example on Friedrich Hirzebruch. In a second research phase, Kodaira wrote a long series of papers in
Kunihiko_Kodaira
smooth manifolds Riemann–Roch theorem for surfaces Grothendieck–Hirzebruch–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Riemann–Stieltjes integral
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
Mathematics award
2014. Mario J. Micallef; J. Gray. "The work of Jesse Douglas on Minimal Surfaces" (PDF). Wdb.ugr.es. Archived from the original (PDF) on 6 October 2014
Fields_Medal
German mathematician (born 1961)
Kiel, he received his Dr. rer. nat. under the direction of Friedrich Hirzebruch at the University of Bonn in 1989. Göttsche was invited as speaker to
Lothar_Göttsche
On the intersection form of a smooth, closed 4-manifold with a spin structure
seen from Friedrich Hirzebruch's signature theorem. The case d = 4 {\displaystyle d=4} gives back the last example of a K3 surface. Michael Freedman's
Rokhlin's_theorem
Coherent sheaf Invertible sheaf Sheaf cohomology Coherent sheaf cohomology Hirzebruch–Riemann–Roch theorem Grothendieck–Riemann–Roch theorem Coherent duality
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Algebraic structure associated with a topological space
French, note 41, explicitly names Noether as inventing the homology group. Hirzebruch, Friedrich, Emmy Noether and Topology in Teicher 1999, pp. 61–63. Bourbaki
Homology_(mathematics)
Differential operator
eta invariant, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at
Eta_invariant
French mathematician (1928–2014)
algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem, a generalisation of the Hirzebruch–Riemann–Roch theorem proved algebraically;
Alexander_Grothendieck
Characteristic class for real vector bundles
the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem. There is also a quaternionic Pontryagin class, for
Pontryagin_class
Gives general conditions under which sheaf cohomology groups with indices > 0 are zero
with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem. The statement of Kunihiko Kodaira's result is that
Kodaira_vanishing_theorem
German mathematician (born 1938)
Ph.D. in 1966 from the University of Bonn; his advisor was Friedrich Hirzebruch and his thesis was entitled "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten"
Friedhelm_Waldhausen
Version without requiring the smooth manifolds involved to carry a complex structure
Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without
Riemann–Roch theorem for smooth manifolds
Riemann–Roch_theorem_for_smooth_manifolds
Property of an algebraic variety
doi:10.1007/978-1-4757-3849-0. ISBN 978-1-4419-2807-8. S2CID 197660097. Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics
Arithmetic_genus
Theory in algebraic topology
{\displaystyle {H_{2k+1}}(\mathbb {CP} ^{n};\mathbb {Z} )=0.} The Atiyah–Hirzebruch spectral sequence is the analogous method of computing the (co)homology
Cellular_homology
Rational numbers with root 5 added
instance by Rokhsar, Mermin & Wright 1987; Lehrer & Taylor 2009, p. 253. Hirzebruch 1976; Sporn 2021. Dodd 1983, p. 11. Dodd 1983, pp. 28–29. Dodd 1983, pp
Golden_field
Generalizations of codimension-1 subvarieties of algebraic varieties
vector space when X is a projective curve. Successive generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem, give some
Divisor_(algebraic_geometry)
form on the second cohomology is given by b+ = 1 + 2pg. Moreover, by the Hirzebruch signature theorem c12 (X) = 2e + 3σ, where e = c2(X) is the topological
Noether_inequality
Topics referred to by the same term
work of Max's daughter Emmy Noether Noether inequality Special divisor Hirzebruch–Riemann–Roch theorem This disambiguation page lists articles associated
Max_Noether's_theorem
Great circle with a characteristic length
conjecture, according to which the unit hemisphere is the minimum-area surface having the Riemannian circle as its boundary. Dress, Andreas W. M.; Maehara
Metric_circle
eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at
Shimizu_L-function
Type of topological space
compute an extraordinary (co)homology theory for a CW complex, the Atiyah–Hirzebruch spectral sequence is the analogue of cellular homology. Some examples:
CW_complex
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
showing this hypothetical cobordism invalidates certain properties of the Hirzebruch signature theorem. Let B 4 {\displaystyle B^{4}} be the unit ball in R
Exotic_sphere
Mathematics concept
complete intersections using a generating function described by Friedrich Hirzebruch. For a three-dimensional manifold, for example, the Hodge diamond has
Homological_mirror_symmetry
particles". In Yau, Shing-Tung (ed.). Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer. Surveys in Differential Geometry. Vol. 7. Somerville, Massachusetts:
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Concept in differential geometry
{\displaystyle w_{2}(E)} vanishes. This is a result of Armand Borel and Friedrich Hirzebruch. Furthermore, in the case E → M {\displaystyle E\to M} is spin, the number
Spin_structure
Type of mathematical functions
complex manifold M. Riemann–Roch on a Riemann surface for a vector bundle was proved by Weil in 1938. Hirzebruch generalized the theorem to compact complex
Function of several complex variables
Function_of_several_complex_variables
German mathematician
2009, he solved a 55-year-old open problem posed in 1954 by Friedrich Hirzebruch, which asks "which linear combinations of Chern numbers of smooth complex
Dieter_Kotschick
Algebraic variety in a projective space
generalization of the Riemann–Roch theorem to higher dimension is the Hirzebruch–Riemann–Roch theorem, as well as the far-reaching Grothendieck–Riemann–Roch
Projective_variety
Application of K-theory in string theory
cohomology we must quotient by these large gauge transformations. The Atiyah–Hirzebruch spectral sequence constructs twisted K-theory, with a twist given by the
K-theory_(physics)
Grauert–Riemenschneider vanishing theorem (algebraic geometry) Grothendieck–Hirzebruch–Riemann–Roch theorem (algebraic geometry) Grothendieck's connectedness
List_of_theorems
German mathematician (1894–1971)
Beno Eckmann Hans Freudenthal Alfred Frölicher Werner Gysin Friedrich Hirzebruch Michel Kervaire Willi Rinow Hans Samelson Ernst Specker Eduard Stiefel
Heinz_Hopf
Number, approximately 3.14
"Ch. 5 What is π?". In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold
Pi
really any official status. Although, associated ruled surfaces were classical objects. See Hirzebruch–Riemann–Roch theorem for his result. He was seeking
Vector bundles on algebraic curves
Vector_bundles_on_algebraic_curves
Subject area in mathematics
theorem was generalized by Friedrich Hirzebruch to all algebraic varieties. In Hirzebruch's formulation, the Hirzebruch–Riemann–Roch theorem, the theorem
Algebraic_K-theory
Field of knowledge
from the original on 2022-11-17. Retrieved 2022-11-17. Chern, S. S.; Hirzebruch, F. (2000). Wolf Prize in Mathematics. doi:10.1142/4149. ISBN 978-981-02-3945-9
Mathematics
Graduate-level textbooks in mathematics
Anderson 1972-03-21 308 978-0691080871 70 Prospects in Mathematics. F. Hirzebruch, Lars Hörmander, John Milnor, Jean-Pierre Serre, I. M. Singer 1971-11-21
Annals_of_Mathematics_Studies
One of six awards by the Wolf Foundation
contributions to many areas of analysis and applied mathematics. 1988 Friedrich Hirzebruch Germany for outstanding work combining topology, algebraic geometry and
Wolf_Prize_in_Mathematics
German mathematician (born 1952)
Gritsenko, Valery; Hulek, Klaus; Sankaran, G. K. (27 September 2006). "The Hirzebruch-Mumford volume for the orthogonal group and applications". arXiv:math/0512595
Klaus_Hulek
Complex-differentiable part of a Maass wave function
MR 0969247, S2CID 121384412 Hirzebruch, Friedrich; Zagier, Don (1976), "Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus"
Mock_modular_form
Chinese-American mathematician and poet
Stanley of the Massachusetts Institute of Technology (1997), and Friedrich Hirzebruch of the Max Planck Institute for Mathematics in Bonn (1998). Jean-Pierre
Shiing-Shen_Chern
Topological spaces whose union is a boundary
developments in topology in the 1950s and early 1960s, in particular in the Hirzebruch–Riemann–Roch theorem, and in the first proofs of the Atiyah–Singer index
Cobordism
Mathematics study in geometry
stability condition Homological mirror symmetry Shklyarov, D. (2013). "Hirzebruch-Riemann-Roch-type formula for DG algebras". Proceedings of the London
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
Rational number sequence
{Numerator} \left({\frac {B_{4n}}{4n}}\right).} The Hirzebruch signature theorem for the L genus of a smooth oriented closed manifold
Bernoulli_number
Branch of geometry
der Konvexgeometrie und der Geometrie der Zahlen", in Fischer, Gerd; Hirzebruch, Friedrich; Scharlau, Winfried; Törnig, Willi (eds.), Ein Jahrhundert
Convex_geometry
Every polynomial has a real or complex root
Fundamental Theorem of Algebra", in Ebbinghaus, Heinz-Dieter; Hermes, Hans; Hirzebruch, Friedrich (eds.), Numbers, Graduate Texts in Mathematics 123, Berlin:
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Concept in algebraic geometry
E, according to the Riemann–Roch theorem and its generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem. For example
Coherent_sheaf_cohomology
Four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations
and various characteristic classes, such as Euler characteristic, the Hirzebruch signature (Pontryagin class), the Rarita–Schwinger index (spin-3/2 index)
Gravitational_instanton
American mathematician (1904–1988)
2013. Hans Lewy at the Mathematics Genealogy Project Chern, Shiing-Shen; Hirzebruch, Friedrich, eds. (2001). ""Hans Lewy" prepared by S. Hildebrandt". Wolf
Hans_Lewy
German mathematician and math historian
besides Purkert, Egbert Brieskorn (the project initiator), Friedrich Hirzebruch, Reinhold Remmert, and Erhard Scholz. There is a first edition done by
Walter_Purkert
method) Friedrich Hirzebruch, The topology of normal singularities of an algebraic surface (normal singularity, algebraic surfaces) Jean-Louis Koszul
Séminaire Nicolas Bourbaki (1960–1969)
Séminaire_Nicolas_Bourbaki_(1960–1969)
equivalence classes. These correspond to the cohomology classes in the Atiyah Hirzebruch Spectral Sequence construction of twisted K-theory, which are only defined
Ramond–Ramond_field
Algebraic structure
readily computable: there is a combinatorial formula found by Friedrich Hirzebruch. The machinery based on the notions of Hodge structure and mixed Hodge
Hodge_structure
own result. Grothendieck reinterpreted both sides of the formula that Hirzebruch proved in 1953 in the framework of morphisms between varieties, resulting
List of publications in mathematics
List_of_publications_in_mathematics
Mathematical theory
Physicist typically want to calculate twisted K-theory using the Atiyah–Hirzebruch spectral sequence. The idea is that one begins with all of the even or
Twisted_K-theory
History of maths
relative point of view, S-schemes. 1957 Alexander Grothendieck Grothendieck–Hirzebruch–Riemann–Roch theorem for smooth schemes; the proof introduces K-theory
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
field theory) Henri Cartan, Fonctions et variétés algebroïdes, d'après F. Hirzebruch (algebroid functions, algebroid varieties) Robert Pallu de la Barrière
Séminaire Nicolas Bourbaki (1950–1959)
Séminaire_Nicolas_Bourbaki_(1950–1959)
Generalization of a differentiable manifold
these homology theories evaluated at a point, the Euler homology and the Hirzebruch homology respectively. Suppose, one has a closed embedding i : N ↪ M {\displaystyle
Stratifold
Endowed chair at Trinity College Dublin
Discrete symmetry properties and elementary particles 1970–1971: Friedrich Hirzebruch (1927–2012) Some relations between topology and number theory 1971–1972:
Donegall Lectureship at Trinity College Dublin
Donegall_Lectureship_at_Trinity_College_Dublin
American annual mathematics conference
related fields. Marcel Berger Pat Eberlein Jost Eschenburg Friedrich Hirzebruch Blaine Lawson Leon Simon Scott Wolpert Deane Yang Uwe Abresch, Explicit
Geometry_Festival
Alan Thorne, Australian anthropologist (b. 1939). 27 May – Friedrich Hirzebruch, German mathematician (b. 1927). 30 May – Sir Andrew Huxley, British physiologist
2012_in_science
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