By James A. Carlson, C. Herbert Clemens, David R. Morrison

Within the overdue Nineteen Sixties and early Nineteen Seventies, Phillip Griffiths and his collaborators undertook a research of interval mappings and version of Hodge constitution. The motivating difficulties, which based at the figuring out of algebraic forms and the algebraic cycles on them, got here from algebraic geometry. even if, the techiques used have been transcendental in nature, drawing seriously on either Lie idea and hermitian differential geometry. Promising methods have been formulated to basic questions within the thought of algebraic curves, moduli thought, and the deep interplay among Hodge conception and algebraic cyles. quick development on many fronts was once made within the Seventies and Eighties, together with the invention of significant connections to different fields, together with Nevanlinna conception, integrable platforms, rational homotopy idea, harmonic mappings, intersection cohomology, and superstring conception. This quantity comprises 13 papers awarded throughout the Symposium on complicated Geometry and Lie idea held in Sundance, Utah in could 1989. The symposium used to be designed to check 20 years of interplay among those fields, targeting their hyperlinks with Hodge conception. The organizers felt that the time used to be correct to envision once more the big problems with knowing the moduli and cycle idea of higher-dimensional kinds, which used to be the place to begin of those advancements. The breadth of this number of papers shows the ongoing progress and power of this sector of study. a number of survey papers are incorporated, which may still make the publication a invaluable source for graduate scholars and different researchers who desire to know about the sector. With contributions from a few of the field's best researchers, this quantity testifies to the breadth and power of this region of analysis

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65) on an arbitrary 1-form µ ∈ 1 (G). The left-hand side gives J ∗ µ, [ρM (v), π (α)] = d(J ∗ µ)(ρM (v), π (α)) + LρM (v) J ∗ µ, π (α) − Lπ (α) J ∗ µ, ρM (v) ∗ ∗ (α)) − LρM (v) µ, (σ ∨ )∗ ρM (α) = −(dµ)(ρ(v), (σ ∨ )∗ ρM + L(σ ∨ )∗ ρM∗ (α) µ, ρ(v) . 67) Evaluating µ on the right-hand side, we get − LρM (v) (α), ρM σ ∨ µ = −LρM (v) α, ρM σ ∨ µ + α, [ρM (v), ρM σ ∨ µ] . Now, using [ρM (v), ρM (v)] ˜ = ρM ([v, v]) ˜ + ρM LρM (v) (v) ˜ for v˜ = σ ∨ µ ∈ ∞ C (M, g), we get ∗ ∗ ∗ −LρM (v) ρM (α), σ ∨ µ + ρM (α), [v, σ ∨ µ] + ρM (α), LρM (v) (σ ∨ µ) .

For any manifold M equipped with an inﬁnitesimal action ρM : g −→ T M, and any g-equivariant map J : M −→ G, the operator 1 C = 1 − ρM ρ ∨ (dJ ) : T M −→ T M 4 and its dual C ∗ : T ∗ M −→ T ∗ M satisfy the formulas ρM σ ∨ σ = CρM , and J ∗ σ σ ∨ = C ∗ J ∗ . 16 follows from the next two propositions, each one describing explicitly one direction of the asserted one-to-one correspondence. 19. Let M be a quasi-Poisson g-manifold, and let A = T ∗ M ⊕ g be its associated Lie algebroid, with anchor r.

The second identity is immediate from the ﬁrst and the last ones. To prove the last identity, we evaluate j (U (α, v)). The ﬁrst component gives 1 1 ∗ −J ∗ − (ρ ∨ )∗ ρM (α) + σ (v) = −J ∗ σ (v) + α − 1 − (ρM ρ ∨ J )∗ α 4 4 ∗ ∗ = −(J σ (v) + C (α)) + α. 59) The second component is 1 ∗ σ ∨ − (ρ ∨ )∗ ρM α + σ (v) . 22). 60) is 1 v − ρ ∨ dJ (ρM (v) + π (α)). , j ◦ U + i ◦ (r, s) = Id. 19. 20. Let J : (M, L) → (G, LG ) be a Dirac realization. Identifying LG with g G, we know that there is an induced action of g on M, denoted by ρM .