specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape...

Lefschetz pencil of hyperplane sections is a more subtle system than a Morse function because hyperplanes intersect each other). The Picard–Lefschetz...

collectively as Bertini's theorem. The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because...

⌋ ) . {\displaystyle {\mathcal {O}}(\lfloor D\rfloor ).} The Lefschetz hyperplane theorem implies that for a smooth complex projective variety X of dimension...

Hironaka theorem (algebraic geometry) Hodge index theorem (algebraic surfaces) Katz–Lang finiteness theorem (number theory) Lefschetz hyperplane theorem (algebraic...

Nakano vanishing theorems, the Lefschetz hyperplane theorem, Hard Lefschetz theorem, Hodge-Riemann bilinear relations, and Hodge index theorem. On a Riemannian...

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...

such as the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations, and the Hodge index theorem. They are also...

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...

{\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...

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...

\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...

building on Hodge theory. The results include the Lefschetz hyperplane theorem, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations. Many...

vanishing theorem Lefschetz hyperplane theorem: an ample divisor in a complex projective variety X is topologically similar to X. Hartshorne (1977), Theorem II...

and MacPherson. The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map f...

projective spaces Adequate equivalence relation Hilbert scheme Lefschetz hyperplane theorem Minimal model program Kollár & Moduli, Ch I. Shafarevich, Igor...

inverse Parallelizable manifold Thom's and Bott's proofs of the Lefschetz hyperplane theorem Atiyah, Michael (2007). "Raoul Harry Bott. 24 September 1923...

{\displaystyle n} . Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69 (3): 713–717...

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...

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...

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...

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...

\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...

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...

projective space Plane at infinity, hyperplane at infinity Projective frame Projective transformation Fundamental theorem of projective geometry Duality (projective...

analogues of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic...

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...

has an associated long exact sequence in cohomology. From the Lefschetz hyperplane theorem there is only one interesting cohomology group of X {\displaystyle...

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)...

represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been primary tools of mathematical...