Analytic inversion of adjunction: L^2 extension theorems with gain

McNeal, Jeffery D.; Varolin, Dror

doi:10.5802/aif.2273

Cited by 56 publications

(40 citation statements)

References 11 publications

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“…We refer to [34], [35] and [24] for the details concerning the regularization process. Remark also that the only additional conditions needed in order to allow singular metrics are the integrability, and the generic finiteness of the restriction to the manifold we want to extend our form.…”

Section: Theorem ([23])mentioning

confidence: 99%

Bergman kernels and the pseudoeffectivity of relative canonical bundles

Берндтссон

Păun²

2008

Duke Math. J.

174

202

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Abstract. The main result of the present article is a (practically optimal) criterion for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. Our theorem has several applications in algebraic geometry; to start with, we obtain the natural analytic generalization of some semipositivity results due to E. Viehweg and F. Campana. As a byproduct, we give a simple and direct proof of a recent result due to C. Hacon-J. McKernan and S. Takayama concerning the extension of twisted pluricanonical forms. More applications will be offered in the sequel of this article. §0 IntroductionIn this article our primary goal is to establish some positivity results concerning the twisted relative canonical bundle of projective morphisms.Let X and Y be non-singular projective manifolds, and let p : X → Y be a surjective projective map, whose relative dimension is equal to n. Consider also a line bundle L over X, endowed with a -possibly singular-metric h = e −φ , such that the curvature current is semipositive. We denote by I(h) the multiplier ideal sheaf of h (see e.g. [10], [21], [25]). Let X y be the fiber of p over a point y ∈ Y , such that y is not a critical value of p. We also assume at first that the restriction of the metric φ to X y is not identically −∞. Under these circumstances, the space of (n, 0) forms L-valued on X y which belong to the multiplier ideal sheaf of the restriction of the metric h is endowed with a natural L 2 -metric as follows(we use the standard abuse of notation in the relation above). Let us consider an orthonormal basis (uRecall that the bundles K X y and K X/Y |X y are isomorphic. Via this identification (which will be detailed in the paragraph 1) the sections above can be used to define 2 Bergman kernels and the pseudoeffectivity of relative canonical bundles a metric on the bundle K X/Y + L restricted to the fiber X y , called the Bergman kernel metric. This definition immediately extends also to fibers such that the metric φ is identically equal to −∞ on the fiber. In this case the Bergman kernel vanishes identically on the fiber, and the Bergman kernel metric is also equal to −∞ there. LetThen we have the next result, which gives a pseudoeffectivity criterion for the bundle0.1 Theorem. Let p : X → Y be a surjective projective map between smooth manifolds, and let (L, h) be a holomorphic line bundle endowed with a metric h such that: Then the relative Bergman kernel metric of the bundleIt has semipositive curvature current and extends across X \ X 0 to a metric with semipositive curvature current on all of X.Several versions of the theorem above were established by the first author in his series of articles on the plurisubharmonic variation of the Bergman kernels (see [1], [2], [3] and also [22] for the first results in this direction). Let us point out the main improvements we have got in the present article. In the first place, we allow the metric h to be singular. Secondly, the map p is not supposed to be a smooth fibration-this will be crucial for t...

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Section: Theorem ([23])mentioning

confidence: 99%

Bergman kernels and the pseudoeffectivity of relative canonical bundles

Берндтссон

Păun²

2008

Duke Math. J.

174

202

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“…The big breakthrough came recently with a very short proof by Chen [40] who was the first one to succeed in deducing the Ohsawa-Takegoshi theorem directly from Hörmander's estimate. In fact he proved even a slightly more general result, obtained earlier by McNeal and Varolin [91] with more complicated methods: Theorem 3.2 Assume that ⊂ C n−1 × is pseudoconvex and let H := {z n = 0}. Then for any ϕ ∈ P S H( ) and f holomorphic in := ∩ H there exists a holomorphic extension F of f in satisfying…”

Section: Ohsawa-takegoshi Extension Theoremmentioning

confidence: 98%

Cauchy–Riemann meet Monge–Ampère

Błocki

2014

Bull. Math. Sci.

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“…Then there exists a neighborhood U ⊂ W of p and positive constants C such that (35) holds with appearing in (iii) above.…”

Section: Theorem 74 If Satisfies Propertyp Then N Is Compactmentioning

confidence: 99%

“…In [35], the authors introduced an approach to L 2 extension that encompassed all of the gain-type results discussed so far. At the heart of the result is the notion of denominators, which we now present.…”

Section: Theory Of Denominators and A General L 2 Extension Theorem mentioning

confidence: 99%

$$L^2$$ L 2 estimates for the $$\bar{\partial }$$ ∂ ¯ operator

McNeal

Varolin

2015

Bull. Math. Sci.

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We present the theory of twisted L 2 estimates for the Cauchy-Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of Ohsawa-Takegoshi type, size estimates on the Bergman kernel, quantitative information on the classical invariant metrics of Kobayshi, Caratheodory, and Bergman, and sub elliptic estimates on the∂-Neumann problem. We endeavor to explain the flexibility inherent to the twisted method, through examples and new computations, in order to suggest further applications.

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Analytic inversion of adjunction: L^2 extension theorems with gain

Cited by 56 publications

References 11 publications

Bergman kernels and the pseudoeffectivity of relative canonical bundles

Bergman kernels and the pseudoeffectivity of relative canonical bundles

Cauchy–Riemann meet Monge–Ampère

$$L^2$$ L 2 estimates for the $$\bar{\partial }$$ ∂ ¯ operator

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