Lefschetz package

In mathematics , the Lefschetz package (also Hodge-Lefschetz package or Kähler package ) is an abstract principle that can be applied in completely different areas of mathematics and enables deep-seated conjectures to be proven in each of these areas.

Abstract definition of a Lefschetz package

A mathematical object a is graduated algebra with associated, one that for a whole number (the "dimension" of the object ) and each integer having: ${\displaystyle X}$ ${\displaystyle A^{*}(X)}$ ${\displaystyle A^{0}(X)=\mathbb {R} }$ ${\displaystyle d}$ ${\displaystyle X}$ ${\displaystyle k}$

• Poincaré duality : a vector space isomorphism

${\displaystyle PD\colon A^{k}(X)\to A^{d-k}(X)^{*}}$,

where denotes the vector space to be dual . ${\displaystyle A^{*}}$ ${\displaystyle A}$

• Lefschetz heavy theorem : a linear mapping such that ${\displaystyle L\colon A^{k}(X)\to A^{k+1}(X)}$

${\displaystyle HL:=L^{d-2k}\colon A^{k}(X)\to A^{d-k}(X)}$

is a vector space isomorphism.

• Hodge-Riemann relations : the pairing given by PD and HL

${\displaystyle A^{k}(X)\times A^{k}(X)\to A^{d}(X)=\mathbb {R} }$

is symmetrical as well as positive definite on the core of . ${\displaystyle L^{d-2k+1}}$

Examples

Kähler geometry

Let be a closed Kähler manifold with Kähler form . Then through the classic Poincaré duality and through ${\displaystyle M}$ ${\displaystyle \omega }$

${\displaystyle L(.):=\omega \wedge .}$

induced linear mapping of a Lefschetz packet defined on the De Rham cohomology . This has numerous applications in the theory of Kähler manifolds, including the Hodge index theorem and the construction and properties of period maps . ${\displaystyle L}$ ${\displaystyle H_{dR}^{*}(M)}$

Furthermore, the restriction of also defines a Lefschetz package on the Dolbeault cohomology . More generally, one can look at their cut cohomology for complex projective varieties and then also obtain a Lefschetz decomposition . ${\displaystyle L}$ ${\displaystyle \textstyle \bigoplus _{k}H^{k,k}(M)}$ ${\displaystyle X}$ ${\displaystyle \bigoplus _{k}H^{k,k}(X)}$

Algebraic Geometry

Be an algebraic variety . Grothendieck's standard conjectures state that one has a Lefschetz packet on the vector space of the algebraic cycle moduli of homological equivalence. You are unproven. The Weil conjectures, proven by Deligne with other methods, follow from the standard conjectures . ${\displaystyle X}$

Polytopes and triangulated spheres

The combinatorial cut cohomology of a convex polytope has a Lefschetz package. With the difficult Lefschetz theorem, Stanley proved the g-conjecture for simplicial polytopes. Kalle Karu extended this to general polytopes, and Adiprasito showed the g-conjecture for triangulated spheres. The Alexandrov-Fennel inequality follows from the Hodge-Riemann relations .

Representation theory

Soergel bimodules have a Lefschetz package. From this follows the positivity of the coefficients of the Kazhdan-Lusztig polynomials as well as an algebraic proof ( Geordie Williamson , Ben Elias ) of the Kazhdan-Lusztig conjecture , a character formula for representations, previously proven by Beilinson - Bernstein , Brylinski - Kashiwara and later Soergel with other methods highest weight .

Matroids

The Chow Ring of a matroid has a Lefschetz package. From the Hodge-Riemann relations it follows that the chromatic polynomial of the matroid is log-concave and thus unimodal . For a sequence of real numbers , log-concave means that the following applies to the terms of the sequence , and unimodular means that there is a sequence term such that (the sequence consists of n sequence terms) , i.e. it has a maximum and is otherwise monotonically falling on one side and on the other monotonously increasing. These properties hold particularly true for the coefficients of the chromatic polynomial of graphs , a conjecture by Ronald C. Read that June Huh proved before proving the more general case of the matroids. ${\displaystyle a_{i}}$ ${\displaystyle a_{i+1}a_{i-1}\leq a_{i}^{2}}$ ${\displaystyle a_{k}}$ ${\displaystyle a_{1}\leq \cdots \leq a_{k-1}\leq a_{k}\geq a_{k+1}\geq \cdots \geq a_{n}}$

literature

• Claire Voisin : Hodge theory and the topology of compact Kähler and complex projective manifolds. on-line
• June Huh: Tropical geometry of matroids. on-line
• June Huh: Combinatorial applications of the Hodge-Riemann relations, Proc. ICM 2018, Arxiv

Individual evidence

1. ↑ A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers , Asterisque (1982)
2. ^ R. Stanley: The number of faces of a simplicial convex polytope , Adv. Math. 35, 236-238 (1980)
3. ↑ K. Karu: Hard Lefschetz theorem for nonrational polytopes , Invent. Math. 157, 419-447 (2004)
4. ↑ Gil Kalai: Amazing: Karim Adiprasito proved the g-conjecture for spheres! In: Combinatorics and more. December 25, 2018, accessed January 26, 2019 . 
5. ↑ B. Elias, G. Williamson: The Hodge theory of Soergel bimodules , Ann. Math. 180, 1089-1136 (2014)
6. ↑ K. Adiprasito, J. Huh, E. Katz: Hodge theory for combinatorial geometries , Ann. Math. 188 (2018)