Results on Griffiths positivity, glued bundles, symmetric-power filtrations and plurisubharmonic limits

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Normal frames — Lemma 1.1. For a smooth Hermitian metric on a holomorphic vector bundle and a point, there is a holomorphic frame in which the metric matrix is the identity at the point and has vanishing first derivatives there.
• The dual norm criterion — Proposition 3.1. If a smooth Hermitian metric on a holomorphic vector bundle is Griffiths-semipositive, the squared norm of the dual metric is plurisubharmonic on the total space of the dual bundle.
• Ample bundles without Griffiths-positive metrics — Theorem 5.1. There is a rank-two bundle G on P^1 x P^1 such that every f_m^*G tensor O(1,1) is ample, and for all large m none of them carries a smooth strictly Griffiths-positive Hermitian metric: Griffiths' conjecture fails in rank two on the quadric surface.
• Two elliptic matrices with a hyperbolic product — Lemma 1.1. Explicit A, B in SL_2(C) of order six, with A^-1 B of order eight, acting on P^1 with order three and without a common eigenline, whose product has real eigenvalues lambda > 1 > 1/lambda.
• Three graphs on the quadric — Lemma 2.1. The graphs of I, A and B in P^1 x P^1 form a divisor of bidegree (3,3) with exactly six transverse nodes and no triple point.
• A bundle glued along three graphs — Proposition 3.1. The 24th tensor powers of the lifts glue to an isomorphism of O(-24,0) and O(0,-24) along the graphs; the pairs of sections of O(24,0) + O(0,24) that agree under the dual identification form a rank-two bundle G with two quotient sequences, whose dual line subbundles span off the graphs and are glued along them.
• Symmetric powers of an extension — Lemma 1.1. For an extension 0 -> K -> V -> L -> 0 with line bundles K and L, multiplication gives exact sequences 0 -> K tensor Sym^(r-1) V -> Sym^r V -> L^r -> 0.
• A filtration by line bundles — Lemma 2.1. For even n, Sym^n f_m^*G has a filtration by subbundles with n + 1 line bundle quotients, obtained by using the two quotient sequences alternately.
• Ampleness of the power family — Theorem 4.1. Every E_m = f_m^*G tensor O(1,1) is ample; O_{P(E_m)}(n) is very ample for every even n with n - 1 >= 3m.
• Continuous functions plurisubharmonic off a hypersurface — Lemma 1.1. A continuous function that is plurisubharmonic off the zero set of a holomorphic function not identically zero on any component is plurisubharmonic.
• A quadratic limit of dual norms — Proposition 5.1. If f_m^*G tensor O(1,1) carries a strictly Griffiths-positive metric for infinitely many m, there is a nonzero nonnegative plurisubharmonic function on the total space of the dual of G that is a Hermitian quadratic form on every fiber.
• The measure of a squared seminorm — Lemma 2.1. A nonzero plurisubharmonic squared seminorm on a line bundle over a Riemann surface has a subharmonic, locally integrable logarithmic weight whose dd^c is a positive measure, invariant under the automorphisms preserving the seminorm; on O(-b) over P^1 its mass is 2 pi b.
• No common invariant measure — Lemma 3.1. The projective transformations A and B, of order three with a hyperbolic product and without a common fixed point, preserve no nonzero finite positive Borel measure on P^1.
• The obstruction — Proposition 4.1. For the glued bundle G, every finite nonnegative plurisubharmonic function on the total space of the dual of G that is a Hermitian quadratic form on every fiber vanishes identically.

Reading order

Five lessons, to be read in order; the third (ampleness) can be read before the fourth and fifth (the metric obstruction). The theorem is proved in full; plurisubharmonic functions, schemes, projective bundles, cohomology and weak compactness are taken from other courses of the programme and cited where they are used.