Carleman estimates for the laplace-beltrami equation on complex manifolds

Andreotti, Aldo; Vesentini, Edoardo

doi:10.1007/bf02684398

Cited by 304 publications

(252 citation statements)

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“…The basic existence theorem is the following result, which is essentially due to Hörmander [Hö65] and, in a more geometric setting, to Andreotti-Vesentini [AV65].…”

Section: Hörmander's L 2 Estimates and Existence Theoremsmentioning

confidence: 99%

Homological Algebra of Mirror Symmetry

Kontsevich

1995

Proceedings of the International Congress of Mathematicians

884

1,045

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Abstract. One important problem arising in algebraic geometry is the computation of effective bounds for the degree of embeddings in a projective space of given algebraic varieties. This problem is intimately related to the following question: Given a positive (or ample) line bundle L on a projective manifold X, can one compute explicitly an integer m 0 such that mL is very ample for m m 0 ? It turns out that the answer is much easier to obtain in the case of adjoint line bundles 2(K X + mL), for which universal values of m 0 exist. We indicate here how such bounds can be derived by a combination of powerful analytic methods: theory of positive currents and plurisubharmonic functions (Lelong), L 2 estimates for ∂ (Andreotti-Vesentini, Hörmander, Bombieri, Skoda), Nadel vanishing theorem, Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities.

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“…The basic existence theorem is the following result, which is essentially due to Hörmander [Hö65] and, in a more geometric setting, to Andreotti-Vesentini [AV65].…”

Section: Hörmander's L 2 Estimates and Existence Theoremsmentioning

confidence: 99%

Homological Algebra of Mirror Symmetry

Kontsevich

1995

Proceedings of the International Congress of Mathematicians

884

1,045

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“…In [2], A. Andreotti and E. Vesentini established an L 2 4heory on noncompact complex manifolds and showed that the vanishing of Hl~r(X, o) x ®3) is a direct consequence of certain a priori estimate as well as the vanishing of H r (X, 3}. Their approach is of interest since the cohomology vanishing is reduced to the solvability of a ^-equation with uniform estimates related to weighted L 2 -norms of Carleman type.…”

Section: A For Bymentioning

confidence: 99%

“…Let (X, ds 2 ) be a Hermitian manifold of dimension n and (E, ti) a holomorphic Hermitian vector bundle over X. We denote by L?…”

Section: A For Bymentioning

confidence: 99%

A Vanishing Theorem for Proper Direct Images

Ohsawa

1987

Publ. Res. Inst. Math. Sci.

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Let X be a complex analytic space and 3 a coherent analytic sheaf over X. Jf Xis ^-complete, then H r (X, £F), the r-th cohomology of X with coefficients in 3, vanishes for r > q (cf. Andreotti-Grauert [1]). If moreover X is a g-complete manifold of dimension n and EF is locally free, then one has Ho~r(X, o) x ®EF)=Q by Serre's duality, where co x denotes the canonical sheaf of X and Hl~r denotes the cohomology with compact supports.In [2], A. Andreotti and E. Vesentini established an L 2 4heory on noncompact complex manifolds and showed that the vanishing of Hl~r(X, o) x ®3) is a direct consequence of certain a priori estimate as well as the vanishing of H r (X, 3}. Their approach is of interest since the cohomology vanishing is reduced to the solvability of a ^-equation with uniform estimates related to weighted L 2 -norms of Carleman type.The purpose of the present article is to give a relative version of their theory in the following situation.Let /: X-*S be a morphism between complex analytic spaces and £? a coherent analytic sheaf over X. The q-th direct image sheaf R q f*3! is a sheaf over S defined by (R%£F) x : = lim H q (f-\U) y 3*), where U runs through the u neighbourhoods of x. The #-th proper diiect image sheaf R^EF is defined by (R q f&) x :=limH q g(f-l (U\ 3"), where S denotes the family of supports consisting of the subsets of X on which/is proper.A recent result of K. Takegoshi shows that R q f*a) x =Q for q>Q in the above situation, if X is a complex manifold which is bimeromorphically equivalent to a Stein space.

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“…For the proof we refer the reader to Andreotti-Vesentini [1] . (Proposition 5 of p. 93 in the paper.…”

Section: Proposition 2 W P ' Q (3£) Can Be Considered As the Set Of mentioning

confidence: 99%