Invariant Hodge forms and equivariant splittings of algebraic manifolds

ArticleOriginal scientific text

Title

Authors ¹

Department of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

Let T be a complex torus acting holomorphically on a compact algebraic manifold M and let

e v_{*} : π ₁ (T, 1) \to π ₁ (M, x ₀)

be the homomorphism induced by

e v : T ∋ t \mapsto t x ₀ \in M . W e s h o w t \hat{f} or e a c h T - \in v a r i a n t H o d \geq f or m Ω o n M t h e r e i s a h o l o m or ϕ c f i b r a t i o n p : M \to T w h o s e f i b e r s a r e Ω - p e r p e n d i c \underset{̲}{a} r \to t h e or b i t s . U \sin g t h i s w e p r o v e t \hat{M} i s T - \equiv a r i a n t l y b i h o l o m or ϕ c \to T \times \frac{M}{T} if and o n l y if t h e r e i s a \subset g r o u p Δ o f π ₁ (M) and a H o d \geq f or m Ω o n M s u c h t \hat{}

π₁(M) = im ev_∗ × Δ

and

∫_{β×δ} Ω = 0

f or a l l

β ∈ im ev_∗!$! and δ ∈ Δ.

holomorphic action, fibration, Hodge form, equivariant splitting, algebraic manifold

E. Calabi, On Kähler manifolds with vanishing canonical class, in: Algebraic Geometry and Topology, Princeton Univ. Press, 1957, 78-89.
J. B. Carrell, Holomorphically injective complex toral actions, in: Proc. Second Conference on Compact Transformation Groups, Part 2, Lecture Notes in Math. 299, Springer, 1972, 205-236.
J. Matsushima, Holomorphic vector fields and the first Chern class of a Hodge manifold, J. Differential Geom. 3 (1969), 477-480.
D. Mumford, Abelian Varieties, Oxford Univ. Press, Oxford, 1970.
M. Sadowski, Equivariant splittings associated with smooth toral actions, in: Algebraic Topology, Proc., Poznań 1989, Lecture Notes in Math. 1474, Springer, 1991, 183-193.
M. Sadowski, Holomorphic splittings associated with holomorphic complex torus actions, Indag. Math. (N.S.) 5 (1994), 215-219.