Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

Liu, Gang

doi:10.4007/annals.2016.184.3.4

Cited by 26 publications

(41 citation statements)

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“…To prove Claim 4.4, it suffices to prove that .´; 1/ c.n; / > 0 for´2 B.x; R 10 /. The proof is the same as in claim 3:1 in [22]. We skip the details here.…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 83%

“…The proof is the same as in claim 3:1 in [22]. If is very small, we can make K small and (4.9),(4.10) hold.…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 83%

“…First we construct w 0 i for 1 Ä i Ä n 1. The construction is similar to proposition 3:1 in [22]. For completeness, we include the details.…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 99%

“…The argument follows from a slight modification of [22]. Let z be a constant whose value will be determined in the course of the proof, depending only on n; .…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 99%

“…Our strategy in Theorem 1.2 is as follows: by using an argument in [22], we construct holomorphic coordinates on M 1 around points whose tangent cone split off R 2n 2 . Cheeger-Colding and Cheeger-Colding-Tian estimates say the complement has Hausdorff codimension at least 4.…”

Section: Introductionmentioning

confidence: 99%

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Gromov‐Hausdorff Limits of Kähler Manifolds with Bisectional Curvature Lower Bound

Liu

2017

Comm Pure Appl Math

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Given a sequence of complete Kähler manifolds M n i with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of the complex structure of M i . 1.3. A priori, .M 1 ; p 1 / is merely a metric length space. The theorem says that it admits an additional complex analytic structure, which is induced from the limit of holomorphic functions on small balls of M i . Note this is very similar to [13], where holomorphic functions are replaced by holomorphic sections.Remark 1.4. The conclusion of Theorem 1.2 might be surprising at first glance: the singularities of a normal complex analytic variety have real codimension at least 4 while the metric singularities might have codimension 2. To resolve this problem, we actually prove that metric singularities with tangent cones splitting off R 2n 2 are regular in the complex analytic sense. Compare with [13], where it was shown that complex analytic singularities are the same as metric singularities in the Kähler-Einstein case.During the proof of Theorem 1.2, we obtain a topological result for complete Kähler surfaces with positive bisectional curvature: COROLLARY 1.5. Let .M 2 ; p/ be a complete noncompact Kähler surface with positive bisectional curvature and maximal volume growth. Then M is simply connected. Maximal volume growth means vol.B.p; r// cr 4 for some c > 0 and for all r.Remark 1.6. This result is rather weak. However, according to the author's knowledge, it is new. Indeed, there are very few results on topology of complete noncompact Kähler manifolds with positive bisectional curvature, even with the assumption that the manifold has maximal volume growth. In [24], we succeeded in improving the result here. GANG

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“…To prove Claim 4.4, it suffices to prove that .´; 1/ c.n; / > 0 for´2 B.x; R 10 /. The proof is the same as in claim 3:1 in [22]. We skip the details here.…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 83%

“…The proof is the same as in claim 3:1 in [22]. If is very small, we can make K small and (4.9),(4.10) hold.…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 83%

“…First we construct w 0 i for 1 Ä i Ä n 1. The construction is similar to proposition 3:1 in [22]. For completeness, we include the details.…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 99%

“…The argument follows from a slight modification of [22]. Let z be a constant whose value will be determined in the course of the proof, depending only on n; .…”

Section: Construction Of Good Holomorphic Coordinates On Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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