By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

Those chosen papers of S.S. Chern talk about issues reminiscent of indispensable geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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This implies that E EP (L, H), implying that E is polystable. D. To complete the projectivity of the moduli space, we need the following result. (cf [BS] and [F] for proofs, as well as the recent paper [H]). 10. Let PK be a family of semistable principal H-bundles on X × Spec K, or equivalently, if HK denotes the group scheme H × Spec K, a semi-stable HK -bundle PK on XK . Then there exists a ﬁnite extension L/K, with the integral closure B of A in L, such that, PK , after base change to Spec B, extends to a semistable HB -bundle PB on XB .

Hence The B2,K curves lying on K3 surfaces are exceptional from the point of view of 44 Ivona Grzegorczyk and Montserrat Teixidor i Bigas Brill-Noether Theory of rank two and determinant canonical. This is particularly relevant as generic curves on K3’s are well-behaved from the point of view of classical Brill-Noether Theory. Therefore, higher rank Brill-Noether provides a new tool to study the geometry of Mg (see also some very interesting related results in [AN]). 7 Generalized Cliﬀord bounds Cliﬀord’s Theorem for line bundles was generalized to semistable vector bundles by Xiao (see 4).

22 V. 4. It is immediate that the G-action on Q lifts to an action on Q . 3, φx is a proper injection and hence aﬃne. One knows that f is aﬃne (with ﬁbres G/H). Hence ψ is a G–equivariant aﬃne morphism. 5. Let (E, s) and (E , s ) be in the same G-orbit of Q . Then we have E E . Identifying E with E, we see that s and s lie in the same orbit of Aut G E on Γ (X, E(G/H)). 13 we see that the reductions s and s give isomorphic H-bundles. 13 we see that (E, s) and (E , s ) lie in the same G-orbit. Consider the G-action on Q with the linearisation induced by the aﬃne G-morphism Q −→ Q.