Lefschetz theorem on hyperplane sections of a book

The exercise shows that in the case of sympletic manifolds, the hard lefschetz theorem is equivalent to non. 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. 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. If a space is 3dimensional then its hyperplanes are the 2dimensional planes, while if the space is 2dimensional, its hyperplanes are the 1dimensional lines. Citeseerx document details isaac councill, lee giles, pradeep teregowda. This extra freedom enables us to relate local and global invariants of the variety. 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. However im struggling a lot with the actual proof of the lefschetz hyperplane theorem in the way it is presented there.

Rather than considering the hyperplane section y alone, he put it into a family of hyperplane. Lee, where the cohomology is replaced by the quantum cohomology. The perverse filtration and the lefschetz hyperplane. 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. Part of the lecture notes in mathematics book series lnm, volume 862. 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. It would be interesting to investigah what other sequences of combinatorial interest arise as betti numbers in intersection. Vanishing theorem, the invariance of hodge numbers under deformations, and the birational invariance of certain hodge numbers. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. Line bundle projective variety abelian variety hyperplane section. 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. On the construction of contact submanifolds with prescribed topology ibort, a.

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. Hyperplane arrangements and lefschetz s hyperplane section theorem yoshinaga, masahiko, kodai mathematical journal, 2007. Pnc be a smooth hypersurface of degree d in projective space. Hodge theory and complex algebraic geometry ii by claire. June huh milnor numbers of projective hypersurfaces.

I am supposed to do a presentation on lefschetz hyperplane section theorem via morse theory following milnors morse theory for my algebraic geometry class. 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. Lefschetz used a monodromy argument, showing that a typical deformation. Whitehead kahler manifold kernel lefschetz lemma linear. The topology of complex projective varieties afler s. There is also a version of the quantum hyperplane section theorem due to y. Filtered geometric lattices and lefschetz section theorems over the. Can anyone please refer me to some other material on this subject. In section 3, we use this family to explore the subtlety of the weak lefschetz property under various hyperplane sections and in arbitrary characteristic. X a hyperplane section, x can be obtained from z by a sequence of deformation retracts and attach. 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. The reader is then introduced to lefschetzs first and second theorems. The hard lefschetz theorem, some history and recent progress.

Concise and authoritative, this monograph is geared toward advanced undergraduate and graduate students. 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. 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. The topology of complex projective varieties after s. The statement fails for d 2 and d 3 and for d3 there are in nitely many noether lefschetz components v.

In mathematics, specifically in algebraic geometry and algebraic topology, the lefschetz. 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. We use morse theory to prove that the lefschetz hyperplane theorem holds for compact smooth delignemumford stacks over the site of complex manifolds. However, can we loose some condition, 1 what if we take a nonhypersurface section, or a. Homology theory on algebraic varieties dover books. Another example of this spectral sequence is crucial to our proof of the noether lefschetz theorem. The lefschetz hyperplane theorem for stacks daniel halpernleistner abstract. A lefschetz hyperplane theorem with an assigned base point. He studied the topology of hyperplane sections of algebraic varieties, yielding the weak lefschetz theorems. Analogues of this theorem have been conjectured for algebraic cycles. Using hard lefschetz theorem we would like to prove the following two propositions. The wellknown lefschetz theorem is generalized to the case of two intersecting varieties. Hyperplane arrangements and lefschetzs hyperplane section theorem masahiko yoshinaga. Lefschetz theory iii the book was and still is considered by the specialists to be opaque.

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. A generalization of the lefschetz hyperplanesection theorem springerlink. In this paper we give a guided tour of the proof of the second lefschetz hyperplane section theorem as presented in 2. We also prove some prelimi nary results suggesting that the lefschetz hyperplane theorem holds.

The lefschetz theorem refers to any of the following statements. Frankel, the lefschetz theorem on hyperplane sections, ann. 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. 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. Homology theory on algebraic varieties sciencedirect. 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. 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. Combinatorial applications of tho herd lefschetz theorem hx, satisfies the hard lefschetz theorem. There is a related deeper theorem, also due to lefschetz, the hard lefschetz theorem.

A type of the lefschetz hyperplane section theorem on. 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. 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. The lefschetz hyperplane theorem for complex projective varieties. 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 geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. The following is a version of the lefschetz theorem for a. Frankelthe second lefschetz theorem on hyperplane sections, global analysis.

A generalization of the lefschetz hyperplanesection theorem. A type of the lefschetz hyperplane section theorem on \q. 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. Our main contribution is to give criteria for the morphism fx.

The second lefschetz theorem on hyperplane sections andreotti, aldo frankel, theodore. The decomposition theorem and the topology of algebraic maps. Lefschetz hyperplane section theorem, or the weak lefschetz theorem. Hyperplane arrangements and lefschetzs hyperplane section. The 2003 second volume of this account of kaehlerian geometry and hodge theory starts with the topology of families of algebraic varieties. Lefschetz hyperplane theorem states that if we take a hypersurface section of a projective manifold, then we have some relations on the homology or homotopy groups of hyperplane and the ambient manifold. Morse theory and the lefschetz theorem on hyperplane. Fd to be an isomorphism, where f is a contravariant functor which is. The lefschetz hyperplane section theorem is a result concerning a topological relationship between an algebraic variety and its generic hyperplane section.

July 14, 2005 abstract the lefschetz hyperplane section theorem asserts that an a. Lefschetz type theorems for algebraic cycles and vector. Kyle hofmann, the lefschetz hyperplane section theorem, pdf. Hyperplane sections and the subtlety of the lefschetz. This expresses part of the topology of a variety in terms of the topology of its hyperplane sections. Lefschetz section theorem in tropical geometry by deriving tropical analogues of. Hence dynkins result may be regarded as a consequence of the hard lefschetz theorem for inter section cohomology. This book is comprised of eight chapters and begins with a. The present writeup is based primarily on my notes used while. 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.