Restriction of Tangent Bundle and Semistability
TL;DR: In this paper, the vector bundle ι*T(G/P)→Z 1 ∩ Z 2 is semistable under the assumption that degree(Z i ) ≥ (m − 1)·index(G /P)/m, i = 1, 2.
Abstract: Let G be a simple linear algebraic group defined over ℂ and P ⊂ G a maximal proper parabolic subgroup such that m: = dim ℂ G/P ≥ 5. Let ι: Z 1 ∩ Z 2↪G/P be a smooth complete intersection such that degree(Z i ) ≥ (m − 1)·index(G/P)/m, i = 1, 2. Then the vector bundle ι*T(G/P) → Z 1 ∩ Z 2 is semistable.
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Book•
01 Jan 1997TL;DR: In this paper, the Grauert-Mullich Theorem is used to define a moduli space for sheaves on K-3 surfaces, and the restriction of sheaves to curves is discussed.
Abstract: Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.
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Journal Article•
TL;DR: Laszlo as mentioned in this paper proved Lefschetz's theorem for both the fundamental group and the Picard group for both groups, and proved the same theorem for the Picard groups as well.
Abstract: New updated edition by Yves Laszlo of the book ``Cohomologie locale des faisceaux coh\'erents et th\'eor\`emes de Lefschetz locaux et globaux (SGA 2)'', Advanced Studies in Pure Mathematics 2, North-Holland Publishing Company - Amsterdam, 1968. Published by the Societe Mathematique de France this http URL Original text also available in the LaTeX file.
Dans cet ouvrage, on montre des th\'eor\`emes d'alg\'ebrisation et de puret\'e qui permettent d'obtenir des th\'eor\`emes de type Lefschetz pour le groupe fondamental ou de Picard.
In this monograph algebraization and purity theorems are proved, providing Lefschetz's theorem for both the fundamental group and the Picard group.
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"Restriction of Tangent Bundle and S..." refers background in this paper
...Hence from a theorem of Flenner, [3], we know that the restriction of T G/P to complete intersections of sufficiently large degrees are semistable....
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...1 may be compared with Flenner’s theorem which in this context says that for any smooth complete intersection Z1 ∩ Z2 ↪→ G/P of two hypersurfaces of degree , the pullback ∗T G/P is semistable if ( +m )− 2 − 1 > degree G/P ·max { m2 − 1 4 1 } where degree G/P = c1 m with being the ample generator of Pic G/P (see [3], [5, Theorem 7....
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