By Tjurin Andrej N.
Tyurin A. accrued works, quantity 1. Vector bundles (free internet model, Goettingen, 2008)(ISBN 9783938616741)
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Additional info for Collected works Vol. 1. Vector bundles
Tata Institute Studies in Mathematics No. 5, Oxford University Press, Bombay and London (1970). MR 44 # 219. S. Narasimhan and S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve . Ann. of Math. (2) 101 (1975), 391–417. MR 52 # 5669. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface. Ann. of Math. (2) 89 (1969), 14–51. MR 39 # 3518. E. Newstead, Stable bundles of rank 2 and odd degree over a curve of genus 2. Topology 7 (1968), 205–215.
Akad. Nauk SSSR Ser. Mat. (2) 34 (1970), 338–365. MR 43 # 282. N. Tyurin, The classiﬁcation of vector bundles over an algebraic curve of arbitrary genus. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 657–688. MR 36 # 166a; English translation in Amer. Math. Soc. Transl. (2) 63 (1967), 245–279. N. Tyurin, Five lectures on three-dimensional varieties. Uspekhi Mat. Nauk (5) 27 (1972), 3–50; English translation in Russian Math. Surveys (5) 27 (1972), 1–53. R. R. G. B. I. G. N. N. Tyurina, Algebraic surfaces, Trudy Mat.
Hence π2 : P (τ ) ✲ P (V0 ) is a locally trivial bundle on the Grassmannians. ) The spectral sequence H i (P (V0 ), Ri π2 (τ )) ⇒ H i (P (τ ), τ ) = H i (Gk (V0 ), τV0 ) degenerates into the isomorphism H i (P (τ ), ττ ) = H i (P (V0 ), h), since by (3) Ri π2 (ττ ) = h ⊕ Ri π2 (OP (τ ) ) = 0, for i > 0 1, for i = 0 , since a Grassmann manifold is regular. 1), 2) and 3) of Proposition 1 follow from this, as do 1), 2) and 3) Proposition 2. Assertions 4) and 5) of Proposition 1 can be restated as follows: H i (Gk (V0 ), ad τV0 = 0 ∀i that is (4) H i (P (τ ), Θπ1 ) = 0.