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Daniel Greb

Publications and source records attributed to Daniel Greb.

41 records · Page 3Linked to original sources
arXiv2009

Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kaehler quotient. Additionally, as a byproduct of our discussion we give an example of a complete Kaehlerian non-projective algebraic surface, which may be of independent interest.

math.AG↗
arXiv2009

Projectivity of analytic Hilbert and Kaehler quotients

We investigate algebraicity properties of quotients of complex spaces by complex reductive Lie groups G. We obtain a projectivity result for compact momentum map quotients of algebraic G-varieties. Furthermore, we prove equivariant versions of Kodaira's Embedding Theorem and Chow's Theorem relative to an analytic Hilbert quotient. Combining these results we derive an equivariant algebraisation theorem for complex spaces with projective quotient.

math.AG↗
arXiv2009

Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties

Given a p-form defined on the smooth locus of a normal variety, and a resolution of singularities, we study the problem of extending the pull-back of the p-form over the exceptional set of the desingularization. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along the exceptional set. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov-Sommese vanishing theorem to log canonical threefold pairs follows.

math.AG↗
arXiv2007

Kaehlerian reduction in steps

We study Hamiltonian actions of compact Lie groups K on Kaehler manifolds which extend to a holomorphic action of the complexified group K^C. For a closed normal subgroup L of K we show that the Kaehlerian reduction with respect to L is a stratified Hamiltonian Kaehler K^C/L^C-space whose Kaehlerian reduction with respect to K/L is naturally isomorphic to the Kaehlerian reduction of the original manifold with respect to K.

math.SG↗