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We introduce mixed twistor D-modules and establish their
fundamental functorial properties. We also prove that they can be
described as the gluing of admissible variations of mixed twistor
structures. In a sense, mixed twistor D-modules can be regarded as
a twistor version of M. Saito's mixed Hodge modules. Alternatively,
they can be viewed as a mixed version of the pure twistor D-modules
studied by C. Sabbah and the author. The theory of mixed twistor
D-modules is one of the ultimate goals in the study suggested by
Simpson's Meta Theorem and it would form a foundation for the Hodge
theory of holonomic D-modules which are not necessarily regular
singular.
In this monograph, we de?ne and investigate an algebro-geometric
analogue of Donaldson invariants by using moduli spaces of
semistable sheaves with arbitrary ranks on a polarized projective
surface. We may expect the existence of interesting "universal
relations among invariants", which would be a natural
generalization of the "wall-crossing formula" and the "Witten
conjecture" for classical Donaldson invariants. Our goal is to
obtain a weaker version of such relations, in other brief words, to
describe a relation as the sum of integrals over the products of m-
uli spaces of objects with lower ranks. Fortunately, according to a
recent excellent work of L. Gottsche, H. Nakajima and K. Yoshioka,
[53], a wall-crossing formula for Donaldson invariants of
projective surfaces can be deduced from such a weaker result in the
rank two case. We hope that our work in this monograph would, at
least tentatively, provides a part of foundation for the further
study on such universal relations. In the rest of this preface, we
would like to explain our motivation and some of important
ingredients of this study. See Introduction for our actual problems
and results. Donaldson Invariants Let us brie?y recall Donaldson
invariants. We refer to [22] for more details and precise. We also
refer to [37], [39], [51] and [53]. LetX be a compact simply con- ?
nected oriented real 4-dimensional C -manifold with a Riemannian
metric g. Let P be a principalSO(3)-bundle on X.
This book studies a class of monopoles defined by certain mild
conditions, called periodic monopoles of generalized
Cherkis-Kapustin (GCK) type. It presents a classification of the
latter in terms of difference modules with parabolic structure,
revealing a kind of Kobayashi-Hitchin correspondence between
differential geometric objects and algebraic objects. It also
clarifies the asymptotic behaviour of these monopoles around
infinity. The theory of periodic monopoles of GCK type has
applications to Yang-Mills theory in differential geometry and to
the study of difference modules in dynamical algebraic geometry. A
complete account of the theory is given, including major
generalizations of results due to Charbonneau, Cherkis, Hurtubise,
Kapustin, and others, and a new and original generalization of the
nonabelian Hodge correspondence first studied by Corlette,
Donaldson, Hitchin and Simpson. This work will be of interest to
graduate students and researchers in differential and algebraic
geometry, as well as in mathematical physics.
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