Lefschetz theorem on hyperplane sections of a book

Analogues of this theorem have been conjectured for algebraic cycles. However, can we loose some condition, 1 what if we take a nonhypersurface section, or a. Annals of mathematics, 171 2010, 208921 the perverse. Line bundle projective variety abelian variety hyperplane section. Lefschetz section theorem in tropical geometry by deriving tropical analogues of. On the construction of contact submanifolds with prescribed topology ibort, a. Vanishing theorem, the invariance of hodge numbers under deformations, and the birational invariance of certain hodge numbers. Specically, if x is a complex projective variety of dimension n embedded in c p n and h is a hyperplane, then the maps i. Algebraic variety of complex dimension in the complex projective space, let be a hyperplane passing through all singular points of if any and let be a hyperplane section of. Well also examine the argument used to prove lefschetz s theorem from a more general point of view due to eliashberg, mentioning contact structures and plurisubharmonicity. A generalization of the lefschetz hyperplanesection theorem springerlink.

Morse theory and the lefschetz theorem on hyperplane. The lefschetz hyperplane theorem for complex projective varieties. X a hyperplane section, x can be obtained from z by a sequence of deformation retracts and attach. The natural map h k y, z h k x, z in singular homology is an isomorphism for k lefschetz hyperplane theorem for stacks daniel halpernleistner abstract. Lefschetz theory iii the book was and still is considered by the specialists to be opaque. The hard lefschetz theorem, some history and recent progress. Can anyone please refer me to some other material on this subject. The lefschetz hyperplane section theorem asserts that a complex affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells.

Using hard lefschetz theorem we would like to prove the following two propositions. It would be interesting to investigah what other sequences of combinatorial interest arise as betti numbers in intersection. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. Filtered geometric lattices and lefschetz section theorems over the. The following is a version of the lefschetz theorem for a. Rather than considering the hyperplane section y alone, he put it into a family of hyperplane. In section 3, we use this family to explore the subtlety of the weak lefschetz property under various hyperplane sections and in arbitrary characteristic. The perverse filtration and the lefschetz hyperplane. Pnc be a smooth hypersurface of degree d in projective space.

I more or less understand the proof, but i am really at a loss what could be good and easily presentable applications of the theorem. Our main contribution is to give criteria for the morphism fx. We will discuss lefschetz pencils and will prove lefschetz s theorem on hyperplane sections without using transcendental methods, which will give us a more or less full understanding of the cohomology of nonsingular complex algebraic varieties up to the middle dimension. He proved theorems on the topology of hyperplane sections of algebraic varieties, which provide a basic inductive tool these are now seen as allied to morse theory, though a lefschetz pencil of hyperplane sections is a more subtle system than a morse function because hyperplanes intersect each other. Hyperplane arrangements and lefschetzs hyperplane section.

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. A lefschetz hyperplane theorem with an assigned base point. Hyperplane sections and the subtlety of the lefschetz. The lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells.

The lefschetz algebra l x of a smooth complex projective variety x is the subal gebra of the cohomology algebra of x generated by divisor classes. Combinatorial applications of tho herd lefschetz theorem hx, satisfies the hard lefschetz theorem. Lefschetz hyperplane section theorem, or the weak lefschetz theorem. A theorem of lefschetz andreotti and frankel af have given a proof l of what is known as the lefschetz theorem on hyperplane sections of a nonsingular projective algebraic variety of complex dimension n. Fd to be an isomorphism, where f is a contravariant functor which is. In mathematics, specifically in algebraic geometry and algebraic topology, the lefschetz. There is a related deeper theorem, also due to lefschetz, the hard lefschetz theorem. Frankel, the lefschetz theorem on hyperplane sections, ann. The weak lefschetz theorem or the lefschetz hyperplane theorem states that for a smooth, projective variety y and a smooth hyperplane section x in y, the restriction map of cohomologies hiy to hix is an isomorphism for i less than dimx, and an injection when i equal to dimx.

Lefschetz used a monodromy argument, showing that a typical deformation. Frankelthe second lefschetz theorem on hyperplane sections, global analysis. The lefschetz hyperplane theorem for stacks daniel halpernleistner abstract. A type of the lefschetz hyperplane section theorem on \q. Another example of this spectral sequence is crucial to our proof of the noether lefschetz theorem. The purpose of this paper is to give an explicit description of attaching maps of these cells for the complement of a complex hyperplane arrangement defined over. The lefschetz hyperplane section theorem is a result concerning a topological relationship between an algebraic variety and its generic hyperplane section. Lee, where the cohomology is replaced by the quantum cohomology. Lefschetz theorem with an assigned base point we may assign a base point when applying lefschetz hyperplane theorem unless our variety has a special geometry with respect to the base point. Let me show how lefschetz directly calculates the homology. 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. This expresses part of the topology of a variety in terms of the topology of its hyperplane sections.

We prove a certain fat hyperplane section weak lefschetz type theorem for etale cohomology of nonprojective varieties, similar to a result of goresky and macpherson over complex numbers. He studied the topology of hyperplane sections of algebraic varieties, yielding the weak lefschetz theorems. We use morse theory to prove that the lefschetz hyperplane theorem holds for compact smooth delignemumford stacks over the site of complex manifolds. A summary of the second lefschetz theorem on hyperplane sections, by aldo andreotti and theodore frankel zachary abel in this paper we give a guided tour of the proof of the second lefschetz hyperplane section theorem as presented in 2. Combinatorial applications of the hard lefschetz theorem. A type of the lefschetz hyperplane section theorem on. This extra freedom enables us to relate local and global invariants of the variety.

The decomposition theorem and the topology of algebraic maps. Part of the lecture notes in mathematics book series lnm, volume 862. In this paper we give a guided tour of the proof of the second lefschetz hyperplane section theorem as presented in 2. Lefschetz hyperplane theorem local study of ordinary double points monodromy hard lefschetz theorem lefschetz hyperplane theorem our goal is to understand the topology of a projective variety by analysing its hyperplane slices. The exercise shows that in the case of sympletic manifolds, the hard lefschetz theorem is equivalent to non. A generalization of the lefschetz hyperplanesection theorem.

Hyperplane arrangements and lefschetz s hyperplane section theorem yoshinaga, masahiko, kodai mathematical journal, 2007. Proofs of the lefschetz theorem on hyperplane sections, the picard lefschetz study of lefschetz pencils, and deligne theorems on the degeneration of the leray spectral sequence and the global invariant cycles follow. Lefschetz 17 precise and at the same time easier to understand it is very convenient to modify blow up x along x to get a new variety y with a map f. Citeseerx document details isaac councill, lee giles, pradeep teregowda. I am supposed to do a presentation on lefschetz hyperplane section theorem via morse theory following milnors morse theory for my algebraic geometry class. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real numbers. There is also a version of the quantum hyperplane section theorem due to y. June huh milnor numbers of projective hypersurfaces. The statement fails for d 2 and d 3 and for d3 there are in nitely many noether lefschetz components v.

Hence dynkins result may be regarded as a consequence of the hard lefschetz theorem for inter section cohomology. Kyle hofmann, the lefschetz hyperplane section theorem, pdf. The present writeup is based primarily on my notes used while. The topology of complex projective varieties afler s. The lefschetz theorem refers to any of the following statements. In section 2 we recall the method of lifting an artinian monomial ideal to a set of points and we introduce the family of artinian monomial ideals that we focus on. Introduction this paper arose from an attempt to understand the extent to which the properties of a variety xare re ected by those of an ample cartier divisor d. Homology theory on algebraic varieties sciencedirect. If a space is 3dimensional then its hyperplanes are the 2dimensional planes, while if the space is 2dimensional, its hyperplanes are the 1dimensional lines. The reader is then introduced to lefschetzs first and second theorems. Whitehead kahler manifold kernel lefschetz lemma linear. Concise and authoritative, this monograph is geared toward advanced undergraduate and graduate students. July 14, 2005 abstract the lefschetz hyperplane section theorem asserts that an a. The 2003 second volume of this account of kaehlerian geometry and hodge theory starts with the topology of families of algebraic varieties.

The second lefschetz theorem on hyperplane sections. Hyperplane arrangements and lefschetzs hyperplane section theorem masahiko yoshinaga. Homology theory on algebraic varieties dover books. Lefschetz s theorem on hyperplane sections relates the topology of a complex projective variety in terms of the topology of its hyperplane sections. The second lefschetz theorem on hyperplane sections andreotti, aldo frankel, theodore. The wellknown lefschetz theorem is generalized to the case of two intersecting varieties. We also prove some prelimi nary results suggesting that the lefschetz hyperplane theorem holds. Hodge theory and complex algebraic geometry ii by claire. The topology of complex projective varieties after s.