32Cxx Analytic spaces
- 32C05 Real-analytic manifolds, real-analytic spaces [See also 14Pxx, 58A07]
- 32C07 Real-analytic sets, complex Nash functions [See also 14P15, 14P20]
- 32C09 Embedding of real analytic manifolds
- 32C11 Complex supergeometry [See also 14A22, 14M30, 58A50]
- 32C15 Complex spaces
- 32C18 Topology of analytic spaces
- 32C20 Normal analytic spaces
- 32C22 Embedding of analytic spaces
- 32C25 Analytic subsets and submanifolds
- 32C30 Integration on analytic sets and spaces, currents (For local theory, see 32A25 or 32A27)
- 32C35 Analytic sheaves and cohomology groups [See also 14Fxx, 18F20, 55N30]
- 32C36 Local cohomology of analytic spaces
- 32C37 Duality theorems
- 32C38 Sheaves of differential operators and their modules, D-modules [See also 14F10, 16S32, 35A27, 58J15]
- 32C55 The Levi problem in complex spaces; generalizations
- 32C81 Applications to physics
- 32C99 None of the above, but in this section
1 search hit
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Irreducible symplectic complex spaces
(2012)
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Tim Kirschner
- In Chapter 1 we define period mappings of Hodge-de Rahm type for certain submersive, yet not necessarily locally topologically trivial, morphisms of complex manifolds. Generalizing Griffiths's theory, we interpret the differential of such period mappings as the composition of the Kodaira-Spencer map and a map derived from the sheaf cohomological cup product and the contraction of vector fields with differential forms.
In Chapter 2 of the text, we consider a submersive morphism $f\colon X\to S$ of complex spaces which is compactified by a proper, flat, and Kähler morphism $\bar f\colon \bar X\to S$. Taking into account the codimension of $\bar X\setminus X$ in $\bar X$, we draw conclusions about the degeneration behavior of the relative Frölicher spectral sequence of the morphism $f$ and about the local freeness of the modules $\mathrm{R}^qf_*(\Omega^p_f)$; our results can be viewed as relative generalizations of a theorem of Takeo Ohsawa.
In our final Chapter 3, we employ the upshots of the preceding two chapters in order to deduce a local Torelli theorem for irreducible symplectic complex spaces. As an application of the local Torelli theorem, we prove that irreducible symplectic complex spaces whose codimension of the singular locus does not deceed $4$ satisfy the so-called Fujiki relation.
