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Authors
Johan Martens
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We give a summary of joint work with Michael Thaddeus that realizes toroidal compactifcations of split reductive groups as moduli spaces of framed bundles on chains of rational curves. We include an extension of this work that covers Artin stacks with good moduli spaces. We discuss, for complex groups, the symplectic counterpart of these compactifications, and conclude with some open problems about the moduli problem concerned.
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Let M be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for a given finite set I of natural numbers m>1 with h(m)>0 a projective compactification M of the reduced scheme underlying M such that the ample invertible sheaf L corresponding to the determinant of the direct image of the m-th power of the relative dualizing sheaf on the moduli stack, has a natural extension L to M. A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases natural means that the pullback of L to a curve C --> M, induced by a family f:X --> C is isomorphic to the determinant of the direct image of the m-th power of the relative dualizing sheaf whenever f is birational to a semi-stable family. Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools, hopefully of interest by itself. In particular we will need a theorem on the flattening of multiplier sheaves in families, on their compatibility with pullbacks and on base change for their direct images, twisted by certain semiample sheaves. Following suggestions of a referee, we reorganized the article, we added several comments explaining the main line of the proof, and we changed notations a little bit.
We show that the Brauer group of any moduli space of stable pairs with fixed determinant over a curve is zero.
In this paper, we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of universal Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the wonderful, or De Concini--Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametrizations. A general outline to construct parametrizations is given and verifications for specific structure groups are carried out.
Let G be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal G-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of G. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple G, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev-Manins spaces of weighted pointed curves and with Kauszs compactification of GL(n).
We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${bf A_n}$ ($nge1$), ${bf C_n}$ ($nge0$) and ${bf E_n}$ ($nge0$).
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