New invariants for complex manifolds and isolated singularities

Du, Rong; Luk, Hing Sun; Yau, Stephen S.-T.

doi:10.4310/cag.2011.v19.n5.a7

Cited by 5 publications

(11 citation statements)

References 15 publications

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“…Take a tubular neighborhood U of A k such that U M . By the proof of Proposition 3.9 in [Du et al 2011], we know that the elements in h.…”

Section: Invariants Of Singularitiesmentioning

confidence: 98%

“…Theorem 2.7 [Du et al 2011]. Let V be a 2-dimensional Stein space with 0 as its only normal singular point with ‫ރ‬ -action.…”

Section: Invariants Of Singularitiesmentioning

confidence: 98%

“…In [Du et al 2011] and [Du and Yau 2012], another two sheaves 1;1 V and 1;1 V were considered: Du and Yau showed that these two new sheaves are coherent and found the relation between 1;1 V (respectively, 1;1 V ) and 2 V (respectively, 2 V ) by short exact sequence as follows:…”

Section: Invariants Of Singularitiesmentioning

confidence: 99%

“…Together with Hing Sun Luk and Stephen Yau, the first author introduced in [Du et al 2011] new biholomorphic invariants that give some measurements for this purpose.…”

Section: Introductionmentioning

confidence: 99%

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New invariants for complex manifolds and rational singularities

Gao

2014

Pacific J. Math.

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Dedicated to Professor Stephen S.-T. Yau on the occasion of his sixtieth birthday.Two new invariants f .1;1/ and g .1;1/ were introduced by Du and Yau for solving the complex Plateau problem. These invariants measure in some sense how far away the complex manifolds are from having global complex coordinates. In this paper, we study these two invariants further for rational surface singularities. We prove that these two invariants never vanish for rational surface singularities, which confirms Yau's conjecture for strict positivity of these two invariants. As an application, we solve regularity problem of the Harvey-Lawson solution to the complex Plateau problem for a strongly pseudoconvex compact rational CR manifold of dimension 3. We also construct resolution manifolds for rational triple points by means of local coordinates and show that f .1;1/ D g .1;1/ D 1 for rational triple points.

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“…Take a tubular neighborhood U of A k such that U M . By the proof of Proposition 3.9 in [Du et al 2011], we know that the elements in h.…”

Section: Invariants Of Singularitiesmentioning

confidence: 98%

“…Theorem 2.7 [Du et al 2011]. Let V be a 2-dimensional Stein space with 0 as its only normal singular point with ‫ރ‬ -action.…”

Section: Invariants Of Singularitiesmentioning

confidence: 98%

Section: Invariants Of Singularitiesmentioning

confidence: 99%

“…Together with Hing Sun Luk and Stephen Yau, the first author introduced in [Du et al 2011] new biholomorphic invariants that give some measurements for this purpose.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

New invariants for complex manifolds and rational singularities

Gao

2014

Pacific J. Math.

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“…Let M be a complex manifold of dimension n. It is a natural question to ask how far away this complex manifold is from having global complex coordinates. Together with Hing Sun Luk and Stephen Yau, the first author introduced in [Du et al 2011] new biholomorphic invariants that give some measurements for this purpose.…”

Section: Introductionmentioning

confidence: 99%

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Almaraz

2014

Pacific J. Math.

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We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresponding Euclidean equations.Supported by CAPES and FAPERJ (Brazil). MSC2010: primary 35J65; secondary 53C21.

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On higher dimensional complex Plateau problem

Gao

Yau

2015

Math. Z.

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Abstract. Let X be a compact connected strongly pseudoconvex CR manifold of real dimension 2n − 1 in C N . It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For n ≥ 3 and N = n + 1, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn-Rossi cohomology groups on X in 1981. For n = 2 and N ≥ n + 1, the first and third authors introduced a new CR invariant g (1,1) (X) of X. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. For n ≥ 3 and N > n + 1, the problem still remains open. In this paper, we generalize the invariant g (1,1) (X) to higher dimension as g (Λ n 1) (X) and show that if g (Λ n 1) (X) = 0, then the interior has at most finite number of rational singularities. In particular, if X is Calabi-Yau of real dimension 5, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization.

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New invariants for complex manifolds and isolated singularities

Cited by 5 publications

References 15 publications

New invariants for complex manifolds and rational singularities

New invariants for complex manifolds and rational singularities

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On higher dimensional complex Plateau problem

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