Abstract
I define and investigate period mappings of “Hodge-de Rham type” for submersive, but possibly nonproper, families \(f: X \rightarrow S\) of complex manifolds. The families f under consideration are—that is remarkable—not assumed locally topologically trivial. I rather assume that the sheaves \(\mathrm{R}^{q}f_{{\ast}}(\varOmega _{f}^{p})\) be locally finite free on S and compatible with base change for (p, q) of a fixed total degree n. Moreover, I assume that the Frölicher spectral sequence of f as well as the Frölicher spectral sequences of the fibers of f degenerate in the entries of total degree n. My main result interprets the differential of the period map in the very fashion that Griffiths (Am J Math 90(3):805–865, 1968) has uncovered for f a proper family of manifolds of Kähler type.
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Notes
- 1.
Observe that the base of the family changes from S to the one-point space {s}.
- 2.
When you consult Griffiths’s source, you will notice several conceptual differences to the text at hand. Most prominently, Griffiths works with de Rham and Dolbeault cohomology where I work with abstract sheaf cohomology. Besides, in his construction of the period mapping, he employs a \(\mathcal{C}^{\infty }\) diffeomorphism \(X_{t} \rightarrow X_{s}\) directly in order to obtain the isomorphism ϕ s, t n.
- 3.
Think about how you would prove it. What theorems do you have to invoke?
- 4.
For small categories (i.e., sets) \(\mathcal{C}\) this definition is an actual definition, in the sense that there is a formula in the language of in the language of Zermelo-Fraenkel set theory expressing it. When \(\mathcal{C}\) is large, however, the definition is rather a “definition scheme”—that is, it becomes an actual definition when spelled out for a particular instance of \(\mathcal{C}\).
- 5.
Note that in order to get a real equality here, as opposed to only a \textquotedblleft canonical isomorphism,\textquotedblright you have to work with the correct sheafification functor.
- 6.
Thus, a right splitting of t is nothing but a right inverse of the morphism \(t_{1,2}: t1 \rightarrow t2\).
- 7.
In detail, what you have to prove is this: when I and J are two presheaves of modules on X, then the composition \(I \bar{\otimes } J \rightarrow I^{\#} \bar{\otimes } J^{\#} \rightarrow I^{\#} \otimes J^{\#}\) is isomorphic to the sheafification of \(I \bar{\otimes } J\), where \(\bar{\otimes }\) denotes the presheaf tensor product.
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Kirschner, T. (2015). Period Mappings for Families of Complex Manifolds. In: Period Mappings with Applications to Symplectic Complex Spaces. Lecture Notes in Mathematics, vol 2140. Springer, Cham. https://doi.org/10.1007/978-3-319-17521-8_1
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