14Mxx Special varieties
- 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F45, 13H10]
- 14M06 Linkage [See also 13C40]
- 14M07 Low codimension problems
- 14M10 Complete intersections [See also 13C40]
- 14M12 Determinantal varieties [See also 13C40]
- 14M15 Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
- 14M17 Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]
- 14M20 Rational and unirational varieties [See also 14E08]
- 14M22 Rationally connected varieties
- 14M25 Toric varieties, Newton polyhedra [See also 52B20]
- 14M27 Compactifications; symmetric and spherical varieties
- 14M30 Supervarieties [See also 32C11, 58A50]
- 14M99 None of the above, but in this section
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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.
