The homology of submanifolds of compact Kähler manifolds

Gordon, Gerald Leonard

doi:10.4310/jdg/1214433985

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“…Then we need only take Xi= Y for i=1 and 2 such that X 1 ~X2 =X, which is always possible to do analytically, and X~ ~ X2 is a projective variety if the X~ are. But then it suffices to find X and Y such that X and Y do not satisfy Ap(X, Y) of Gordon [7]. In Gordon [7], it is shown that …”

Section: 1mentioning

confidence: 99%

“…But then it suffices to find X and Y such that X and Y do not satisfy Ap(X, Y) of Gordon [7]. In Gordon [7], it is shown that …”

Section: 1mentioning

confidence: 99%

“…In this section, we use the results of Section 6 and of Gordon [7] to give some examples of compact varieties which can not be the deformation of non-singular compact Kahler varieties.…”

Section: 1mentioning

confidence: 99%

“…In Section 6 we show using the results of Gordon [7] that if all the components of the proper transform are quasi-Kahler, then G3clmage T,-I,, when we take closed support, (6.2); and hence in the middle dimension the local invariant cycle problem, (2.2.1), is true for any analytic deformation of non-singular surfaces when we take closed support, so long as each of the components of V o is quasi-Kahler (6.3). We also show that for any dimension whatsoever, if the total transform of V 0 has only two components, then necessary and sufficient topological conditions can be given with respect to these components to that the local invarient cycle problem is true, (6.1.4).…”

Section: Section I Introductionmentioning

confidence: 99%

“…Finally, in Section 8, using the results of Section 6 and [7], we give some examples of projective varieties which cannot be embedded in a one-dimensional analytic deformation whose generic fibre is a non-singular compact Kahler manifold, (8.1.2).…”

Section: Section I Introductionmentioning

confidence: 99%

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A geometric study of the monodromy of complex analytic surfaces

Gordon

1977

Invent Math

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In this article we will be interested in studying a complex analytic deformation W over the unit disk whose generic fibre 14, is a manifold of complex dimension n. In particular, we shall study the case n = 2 in detail although the techniques will go through in the more general situation. Furthermore in the n=2 case, the techniques should be applicable to studying the monodromy of schemes using etale cohomology.In Section 2 we shall make some topological reductions using resolution of singularities and semi-stable reduction to study the action of T,-I,, where T, is the monodromy, i.e., the action on homology induced from going once around the unit circle, and I, is induced from the identity. We show after a suitable reduction, in particular T, is unipotent, that the action of T,-I, is the same as a boundary operator, (2.4.1). We also show that to study T, -I,, one must study Ker g,, where g, : V~ ~ V o is the collapsing map, i.e., g is the composition of the inclusion of the generic fibre 1/1 into W and the retraction of W onto the singular fibre V o over the origin.In Section 3 we shall study the Ker g,, (3.4), for t1=2, and show that Kerg, c~Hz(V 0 is generated by 3 subspaces: G~, G2 and G3, (3.4). We conjecture that G1 | G2 | G3 c H2(VO, i.e., there are no relations among them.In Section 4, we show by purely topological results that G 1 =(T,-I,)G2, if we take closed support. This is also true for compact support if the proper transform has only one component, i.e., the original V o is irreducible, (4.4.2).However, if V 0 is not irreducible, it can be false that G1 c Image (T,-I,), (4.4.3).In Section 5 we study G 2. We are able to show that if all of the components of the proper transform of V 0 are quasi-Kahler, i.e., each is bianalytically homeomorphic to a compact Kahler manifold minus at most a proper subvariety, then G2 ~ Image (T, -I,), (5.3).In Section 6 we show using the results of Gordon [7] that if all the components of the proper transform are quasi-Kahler, then G3clmage T,-I,, when we take closed support, (6.2); and hence in the middle dimension the local invariant cycle problem, (2.2.1), is true for any analytic deformation of non-*

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Section: 1mentioning

confidence: 99%

“…But then it suffices to find X and Y such that X and Y do not satisfy Ap(X, Y) of Gordon [7]. In Gordon [7], it is shown that …”

Section: 1mentioning

confidence: 99%

“…In this section, we use the results of Section 6 and of Gordon [7] to give some examples of compact varieties which can not be the deformation of non-singular compact Kahler varieties.…”

Section: 1mentioning

confidence: 99%

Section: Section I Introductionmentioning

confidence: 99%

Section: Section I Introductionmentioning

confidence: 99%

See 3 more Smart Citations