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A Mathematician and His Mathematical Work: Selected Papers by Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

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.

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