A categorification of the quantum lefschetz principle. The lefschetz hyperplane theorem for stacks daniel halpernleistner abstract. Suppose xis a topological space and a x is a subspace. The lefschetz property of coinvariant algebras of complex. The strong lefschetz principle in algebraic geometry. Written with spirit and originality, this is a valuable book for anyone interested in the subject from other than the purely algebraic point of view. The geometry of algebraic topology is so pretty, it would seem a pity to slight it and to miss all the intuition it provides. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Intersect with a generic pencil of hyperplanes lefschetz pencil family of curves, at most nodal, through a nite set of base points. A history of duality in algebraic topology james c.
A1 such that grza has the strong lefschetz property. X a hyperplane section, x can be obtained from z by a sequence of deformation retracts and attach. Strong duality was rst employed by poincar e 1893 in a note in which \poincar e duality was used without proof or formal statement. Considering your choice of tags, its plausible that you actually mean the lefschetz principle. Number theory algebraic geometry topological group lefschetz principle these keywords were added by machine and not by the authors. The strong lefschetz property in codimension two cook, david, journal of commutative algebra, 2014. Stanley in 1977 which connected the strong lefschetz property with diverse combinatorial structures, among them the sperner property of a finite poset, using a mixture of algebraic geometry. The influence of solomon lefschetz 18841972 in geometry and topology 40 years after his death has been very profound. The quantum lefschetz formula explains how virtual fundamental classes or structure sheaves of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section.
Symplectic picardlefschetz theory jonny evans eth zurich 17th april 2012. Solomon lefschetz was a towering figure in the mathematical world owing not only to his original contributions but also to his personal influence. The basic idea is that every equation over some algebraically closed field of characteristic only involves finitely many elements, which generate a subfield isomorphic to a subfield of. As in the case of ample bundles the first chern class of lef line bundles satisfies the hard lefschetz theorem and the hodgeriemann bilinear relations.
An introduction to algebraic geometry and a bridge between its analyticaltopological and algebraical aspects, this book explores fundamental concepts of the general theory of algebraic varieties. Claire voisin, section 6 of hodge theory and complex algebraic geometry i,ii, cambridge stud. The various instances of strong duality poincar e, lefschetz, alexander, spanierwhitehead, pontr. Lefschetz pencils and the symplectic topology of complex. Let y be a smooth projective variety over c, and x be a smooth hypersurface in y. An analogue of the lefschetz formula in abstract algebraic geometry has been. Murre, applications of algebraic ktheory to the theory of algebraic cycles, algebraic geometry, sitges barcelona, 1983, volume 1124 of lecture notes in math. Then a has the weak lefschetz property if there is an element ell of degree 1 such that the multiplication times ell. Let x be an ndimensional complex projective algebraic variety in cp n, and let y be a hyperplane section of x such that u x. Combinatorial applications of the hard lefschetz theorem. In its vaguest form, this principle says that algebraic geometry behaves the same way over, over an arbitrary algebraically closed field of characteristic zero, and over an arbitrary algebraically closed field of sufficiently large. The circle of ideas i will discuss today has its roots in the study of algebraic surfaces by picard and lefschetz. A difficulty a student often faces in a subject as diverse as algebraic geometry is the profusion of crossreferences, and this is one reason for attempting to be selfcontained.
An algebraicgeometry property proven in the context of complex algebraic geometry also holds for any algebraically closed field of characteristic zero. Let xbe an ndimensional nonsingular complex projective variety in pn. Finite geometry and the lefschetz property toshiaki maeno department of mathematics, meijo university. The lefschetz principle was formulated and illustrated the first time in. A third general principle was that this volume should be selfcontained. The weak lefschetz property, monomial ideals, and lozenges cook ii, david and nagel, uwe, illinois journal of mathematics, 2011. Ravindra, lefschetz theorems for torsion algebraic cycles in codimensions 2, advances in mathematics 316 2017. Algebraic geometry by solomon lefschetz, paperback. In the twentieth century, the lefschetz principle, named for solomon lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field k of characteristic 0, by treating k as if it were the complex number.
Lefschetz, manifolds with a boundary and their transformations trans. Lefschetz properties of graded algebras the background for the name is the hard lefschetz theorem. The strong lefschetz property and the schurweyl duality. Nonetheless the ideas are central to a modern understanding of symplectic geometry and there will be payo s. Browse other questions tagged algebraicgeometry algebraictopology or ask your own question. The study of lefschetz properties such as the wlp and the slp goes back to the hard lefschetz theorem. In this paper, an alternative proof is presented of the following result on symbolic powers due to ein, lazarsfeld and smith 3 for the affine case over and to hochster and huneke 4 for the general case. The lefschetz theorem refers to any of the following statements. The formulation i will describe was worked out in the late nineties by. For example, as a corollary to tarskis theorem, a strong version of lefschetz principle is obtained. A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of goreskymacpherson. What does the lefschetz principle in algebraic geometry. The coinvariant algebra sn is already proved to have the strong lefschetz property. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of mori theory, fully determines each other.
Lefschetz, algebraic geometry, princeton university press, 1953. The hard lefschetz theorem and the topology of semismall. Inspired by the weak lefschetz principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. The purpose and goal of this workshop is to bring together researchers from these areas in order to share different points of view and to make collaborations possible. A series of modern surveys in mathematics results in mathematics and related areas. The strong lefschetz property for coinvariant rings of. In particular any hard result that would be utilized should be fully proved. This volume contains papers in algebraic geometry and topology contributed by mathematical colleagues of s. Let us consider the strong lefschetz property of the coinvariant algebras of complex re. The first application of modern algebraic techniques to a comprehensive selection of classical geometric problems. Let 8 be a finite subset of r, and for k e r let f 8,k clenots the. The goal has been to feature contemporary research that has developed from the vital basic work of. The influence of solomon lefschetz in geometry and.
We use morse theory to prove that the lefschetz hyperplane theorem holds for compact smooth delignemumford stacks over the site of complex manifolds. On the standard conjecture of lefschetz type for complex. This process is experimental and the keywords may be updated as the learning algorithm improves. The main results obtained in this paper are the following.
Combinatorial applications of the herd lefschetz theorem theorem 2. Lefschetz properties in algebra, geometry and combinatorics. Algebraic topology by lefschetz, solomon, 18841972. Strong lefschetz property under reduction request pdf. Symplectic manifolds and lefschetz pencils algebraic geometry. Lefschetzs influence in mexican mathematics has been even greater. Algebraic geometry and analytic geometry wikipedia. The strong lefschetz principle in algebraic geometry 385 garcia, a weights of weierstrass points in double coverings of curves of genus one or two. Lefschetz decompositions and homological projective. The lefschetz hyperplane theorem for complex projective varieties. More recently these algebraic properties have been studied in various fields of mathematics. A has the strong lefschetz property if and only if there is a linear form z. The main results obtained in this paper are as follows. Introduction the weak and strong lefschetz properties on graded artinian algebras have been an object of study along the last few decades.
At the elementary level, algebraic topology separates naturally into the two broad channels of homology and homotopy. Im trying to understand the proof of the lefschetz hyperplane theorem in milnors book morse theory, page 41 but i cant understand his use of lefschetz duality. The study of the lefschetz properties of commutative algebras in their own right began only relatively recently and seems to go back to a paper of r. Blow up base points lefschetz bration with distinguished sections. In particular well end up seeing a lot more lagrangian submanifolds and symplectomorphisms of projective varieties. He contributed to at least three mathematical fields, and his work reflects throughout deep geometrical intuition and insight.