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Theorem in algebraic geometry
specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape
Lefschetz_hyperplane_theorem
Russian-born American mathematician (1884–1972)
Lefschetz pencil of hyperplane sections is a more subtle system than a Morse function because hyperplanes intersect each other). The Picard–Lefschetz
Solomon_Lefschetz
Belgian mathematician
analogue of the Riemann hypothesis. It also led to a proof of the Lefschetz hyperplane theorem and the old and new estimates of the classical exponential sums
Pierre_Deligne
collectively as Bertini's theorem. The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because
Hyperplane_section
Generalizations of codimension-1 subvarieties of algebraic varieties
⌋ ) . {\displaystyle {\mathcal {O}}(\lfloor D\rfloor ).} The Lefschetz hyperplane theorem implies that for a smooth complex projective variety X of dimension
Divisor_(algebraic_geometry)
and MacPherson. The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map f
Decomposition theorem of Beilinson, Bernstein and Deligne
Decomposition_theorem_of_Beilinson,_Bernstein_and_Deligne
Hironaka theorem (algebraic geometry) Hodge index theorem (algebraic surfaces) Katz–Lang finiteness theorem (number theory) Lefschetz hyperplane theorem (algebraic
List_of_theorems
language, the Picard group is infinite cyclic, other than for a short list of degrees. This is now often called the Noether-Lefschetz theorem. v t e
Max Noether's theorem on curves
Max_Noether's_theorem_on_curves
such as the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations, and the Hodge index theorem. They are also
Kähler_identities
Mathematical manifold theory
building on Hodge theory. The results include the Lefschetz hyperplane theorem, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations. Many
Hodge_theory
Manifold with Riemannian, complex and symplectic structure
Nakano vanishing theorems, the Lefschetz hyperplane theorem, Hard Lefschetz theorem, Hodge-Riemann bilinear relations, and Hodge index theorem. On a Riemannian
Kähler_manifold
Algebraic variety in a projective space
projective spaces Adequate equivalence relation Hilbert scheme Lefschetz hyperplane theorem Minimal model program Kollár & Moduli, Ch I. Shafarevich, Igor
Projective_variety
Generalizes the Kodaira vanishing theorem for ample vector bundle
ISBN 978-3-540-62038-9. S2CID 117583140. Litt, Daniel (2018). "Non-Abelian Lefschetz hyperplane theorems". Journal of Algebraic Geometry. 27 (4): 593–646. arXiv:1601
Le_Potier's_vanishing_theorem
Mathematical sequence
\geq 2} (this is because of the Hurewicz homomorphism and the Lefschetz hyperplane theorem). In this case the local systems R q f ∗ ( Q _ X ) {\displaystyle
Leray_spectral_sequence
Algebraic structure
{\displaystyle H^{1,1}(X)} given by the Lefschetz class [ L ] {\displaystyle [L]} . From the Lefschetz hyperplane theorem and Hodge duality, the rest of the
Hodge_structure
Hungarian-American mathematician (1923-2005)
inverse Parallelizable manifold Thom's and Bott's proofs of the Lefschetz hyperplane theorem Atiyah, Michael (2007). "Raoul Harry Bott. 24 September 1923
Raoul_Bott
Concept in algebraic geometry
vanishing theorem Lefschetz hyperplane theorem: an ample divisor in a complex projective variety X is topologically similar to X. Hartshorne (1977), Theorem II
Ample_line_bundle
non-singular projective surface, and let H be the divisor class on V of a hyperplane section of V in a given projective embedding. Then the intersection H
Hodge_index_theorem
Set of conjectures in algebraic geometry
axioms of a Weil theory is the so-called hard Lefschetz theorem (or axiom): Begin with a fixed smooth hyperplane section W = H ∩ X, where X is a given smooth
Standard conjectures on algebraic cycles
Standard_conjectures_on_algebraic_cycles
Mathematical theorem of complex manifolds
{\displaystyle n} . Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69 (3): 713–717
Andreotti–Frankel_theorem
Characteristic classes of vector bundles
using the definition of the Euler characteristic and using the Lefschetz hyperplane theorem. If X ⊂ P 3 {\displaystyle X\subset \mathbb {P} ^{3}} is a degree
Chern_class
Term in mathematics
\mathbb {CP} ^{n+m}} are the intersection of hyperplane sections, we can use the Lefschetz hyperplane theorem to deduce that H j ( X ) = Z {\displaystyle
Complete_intersection
American mathematician
René Thom, Frankel and Aldo Andreotti gave a new proof of the Lefschetz hyperplane theorem using Morse theory. The crux of the argument is the algebraic
Theodore_Frankel
Concept in algebraic geometry
analogues of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic
Local_cohomology
projective space Plane at infinity, hyperplane at infinity Projective frame Projective transformation Fundamental theorem of projective geometry Duality (projective
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Hungarian and American mathematician and physicist (1903–1957)
represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been primary tools of mathematical
John_von_Neumann
Theory in algebraic geometry
be the inclusion Z ⊂ K. Weak Lefschetz axiom: For any smooth hyperplane section j: W ⊂ X (i.e. W = X ∩ H, H some hyperplane in the ambient projective space)
Weil_cohomology_theory
Mathematical conjecture
structure on H 3 ( X ) {\displaystyle H^{3}(X)} . Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is H 3 ( X ) {\displaystyle
Mirror_symmetry_conjecture
Mathematics glossary
space of formal group laws. Lefschetz 1. Solomon Lefschetz 2. The Lefschetz fixed-point theorem says: given a finite simplicial complex K and its geometric
Glossary of algebraic topology
Glossary_of_algebraic_topology
has an associated long exact sequence in cohomology. From the Lefschetz hyperplane theorem there is only one interesting cohomology group of X {\displaystyle
Jacobian_ideal
equal. Subspace theorem Schmidt's subspace theorem shows that points of small height in projective space lie in a finite number of hyperplanes. A quantitative
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Group of Italian mathematicians who studied birational geometry (c. 1885–1935)
that tendency was Henri Poincaré; during the 1930s it was developed by Lefschetz, Hodge and Todd. The modern synthesis brought together their work, that
Italian school of algebraic geometry
Italian_school_of_algebraic_geometry
Fiber bundle whose fibers are projective spaces
bundle (i.e., the structure sheaf). Then P(E) is a hyperplane in P(E ⊕ 1), called the hyperplane at infinity, and the complement of P(E) can be identified
Projective_bundle
Rational function of the form (az + b)/(cz + d)
path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity)
Möbius_transformation
Algebraic structure used in topology
precursors to cohomology. In the mid-1920s, J. W. Alexander and Solomon Lefschetz founded intersection theory of cycles on manifolds. On a closed oriented
Cohomology
96) pencil A 1-dimensional linear system. See pencil (mathematics) and Lefschetz pencil. pentad A set of 5 points pentahedron A union of 5 planes, in particular
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
American annual mathematics conference
Potential theory for nonlinear PDE's John Pardon (Stanford): Existence of Lefschetz vibrations on Stein/Weinstein domains Raanan Schul (Stony Brook): Qualitative
Geometry_Festival
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LEFSCHETZ HYPERPLANE-THEOREM
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