Grothendieck splitting theorem

Grothendieck's splitting theorem gives a classification of holomorphic vector bundles over a complex projective line . Namely, she claims that every holomorphic vector bundle over $\mathbb {C} \mathrm {P} ^{1}$ ${\ displaystyle \ mathbb {C} \ mathrm {P} ^ {1}}$ ${\ displaystyle \ mathbb {C} \ mathrm {P} ^ {1}}$ is the direct sum of holomorphic 1-dimensional bundles .

Content

History

The theorem is named after Alexander Grothendieck , who proved it in 1957. ^[1] It is equivalent to the theorem previously proved by George Birkhoff in 1913, ^[2] but was already known in 1908 to Josip Plemel ^[3] and in 1905 to David Hilbert . ^[four]

Wording

Grothendieck wording

Every holomorphic vector bundle ${\mathcal {E}}$ ${\ displaystyle {\ mathcal {E}}}$ above $\mathbb {C} \mathrm {P} ^{1}$ ${\ displaystyle \ mathbb {C} \ mathrm {P} ^ {1}}$ holomorphically isomorphic to the direct sum of line bundles:

{\mathcal {E}}\cong {\mathcal {O}}(a_{1})\oplus \cdots \oplus {\mathcal {O}}(a_{n}),

{\ displaystyle {\ mathcal {E}} \ cong {\ mathcal {O}} (a_ {1}) \ oplus \ cdots \ oplus {\ mathcal {O}} (a_ {n}),}

Where ${\mathcal {O}}(a)$ ${\ displaystyle {\ mathcal {O}} (a)}$ denotes a bundle with class Chern $a$ ${\ displaystyle a}$ . Moreover, this representation is unique up to a permutation of the terms.

Birkhoff wording

Invertible matrix $M$ ${\ displaystyle M}$ , each component of which is a Laurent polynomial from $z$ ${\ displaystyle z}$ represented as a work

M=M^{+}M^{0}M^{-}

{\ displaystyle M = M ^ {+} M ^ {0} M ^ {-}}

,

where is the matrix $M^{+}$ ${\ displaystyle M ^ {+}}$ - polynomial from $z$ ${\ displaystyle z}$ , $M^{0}$ ${\ displaystyle M ^ {0}}$ Is the diagonal matrix, and the matrix $M^{-}$ ${\ displaystyle M ^ {-}}$ - polynomial from ${\tfrac {1}{z}}$ ${\ displaystyle {\ tfrac {1} {z}}}$ .

Applications

Grothendieck's splitting theorem is used in the proof of Mikalef and Moore 's sphere theorem for positive complexified curvature in isotropic directions.

Variations and generalizations

The same result holds for algebraic vector bundles over $\mathbb {P} _{k}^{1}$ ${\ displaystyle \ mathbb {P} _ {k} ^ {1}}$ for any field $k$ ${\ displaystyle k}$ . ^[five]

Notes

↑ Grothendieck, Alexander (1957), " Sur la classification des fibrés holomorphes sur la sphère de Riemann ", American Journal of Mathematics Vol . 79: 121 • 138 , DOI 10.2307 / 2372388 .
↑ Birkhoff, George David (1909), " Singular points of ordinary linear differential equations ", Transactions of the American Mathematical Society T. 10 (4): 436–470, ISSN 0002-9947 , DOI 10.2307 / 1988594
↑ Plemelj, J. Riemannsche Funktionenscharen mit gegebener Monodromiegruppe. Monatsh. Math. Phys. 19 (1908), no. 1, 211-245.
↑ Hilbert D. Grundzüge einer allgemeinen theorie der linearen integralgleichungen. vierte mitteilung. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 1906: 157-228.
↑ Hazewinkel, Michiel & Martin, Clyde F. (1982), " A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line ", Journal of Pure and Applied Algebra T. 25 (2): 207–211 , DOI 10.1016 / 0022-4049 (82) 90037-8

Literature

Okonek, C .; Schneider, M. & Spindler, H. (1980), Vector bundles on complex projective spaces , Progress in Mathematics, Birkhäuser .

[1] Grothendieck, Alexander (1957), " Sur la classification des fibrés holomorphes sur la sphère de Riemann ", American Journal of Mathematics Vol . 79: 121 • 138 , DOI 10.2307 / 2372388 .

[2] Birkhoff, George David (1909), " Singular points of ordinary linear differential equations ", Transactions of the American Mathematical Society T. 10 (4): 436–470, ISSN 0002-9947 , DOI 10.2307 / 1988594

[3] Plemelj, J. Riemannsche Funktionenscharen mit gegebener Monodromiegruppe. Monatsh. Math. Phys. 19 (1908), no. 1, 211-245.

[4] Hilbert D. Grundzüge einer allgemeinen theorie der linearen integralgleichungen. vierte mitteilung. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 1906: 157-228.

[5] Hazewinkel, Michiel & Martin, Clyde F. (1982), " A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line ", Journal of Pure and Applied Algebra T. 25 (2): 207–211 , DOI 10.1016 / 0022-4049 (82) 90037-8