Phillip A Griffiths, Mathematiker USA's Topics in transcendental algebraic geometry : (a seminar; PDF

By Phillip A Griffiths, Mathematiker USA

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Let ????1 , . . , ???????? be sections of ????|???? such that, for all ???? ∈ ????, {????1 (????), . . , ???????? (????)} is a basis of the fibre ???????? . We define the connection matrix ???? of ∇ with respect to ????1 , . . , ???????? in the following way: ∇???????? = ∑ ???????? ⊗ ????????,???? (the entries of ???? are 1-forms). ,???? ∇(∑???? ???????? ???????? ) = ∑???? ???????? ⊗ ???????????? + ∑???? ???????? ∇???????? = ∑???? ???????? ⊗ ???????????? + ∑???? ???????? ∑???? ???????? ⊗ ????????,???? = ∑???? ???????? ⊗ (???????????? + ∑???? ???????? ????????,???? ), which can be written for short as ???????? + ????????, Connections where | 33 ????1 .

Call the coordinates in ℙ???? ????????,???? for ???? = 1, . . , ????, ???? = 1, . . , ????. Let ???? be the matrix such that ????????,???? = ????????,???? . We have that ????1 (????) is the image ???? of the Segre embedding (see “Segre embedding”) ℙ????−1 × ℙ????−1 → ℙ???? , (. . , ???????? , . ), (. . , ???????? , . ) ????→ (. . , ???????? ???????? , . ).

Let ℎ???? : C → S be the controvariant functor defined by ℎ???? (????) = ????????????C (????, ????) for any ???? object of C and sending an arrow ???? : ???? → ???? to the arrow ????????????C (????, ????) → ????????????C (????, ????) given by the composition with ????. Sometimes ℎ???? and ℎ???? are denoted respectively by ????????????(????, −) and ????????????( −, ????). Definition. We say that a covariant functor ???? : C → S is representable if it is isomorphic to ℎ???? for some ???? object of C (in this case, we say that ???? represents ????). We say that a controvariant functor ???? : C → S is representable if it is isomorphic to ℎ???? for some ???? object of C (in this case, we say that ???? represents ????).

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