128 research outputs found
Factorization theorems for the representations of the fundamental groups of quasiprojective varieties and some applications
In this paper, using Gromov-Jost-Korevaar-Schoen technique of harmonic maps
to nonpositively curved targets, we study the representations of the
fundamental groups of quasiprojective varieties. As an application of the above
considerations we give a proof of a weak version of the Shafarevich Conjecture.Comment: 50 pages, LaTe
Non-commutative counting invariants and curve complexes
In our previous paper, viewing as a non-commutative curve, where
is the Kronecker quiver with -arrows, we introduced categorical
invariants via counting of non-commutative curves. Roughly, these invariants
are sets of subcategories in a given category and their quotients. The
non-commutative curve-counting invariants are obtained by restricting the
subcategories to be equivalent to . The general definition defines
much larger class of invariants and many of them behave properly with respect
to fully faithful functors. Here, after recalling the definition, we focus on
examples and extend our studies beyond counting. We enrich our invariants with
structures: the inclusion of subcategories makes them partially ordered sets,
and considering semi-orthogonal pairs of subcategories as edges amount to
directed graphs. In addition to computing the non-commutative curve-counting
invariants in for two affine quivers, for and we derive
formulas for counting of the subcategories of type in ,
whereas for the two affine quivers and for we determine and count all
generated by an exceptional collection subcategories. Estimating the numbers
counting non-commutative curves in modulo group action we
prove finiteness and that an exact determining of these numbers leads to
proving (or disproving) of Markov conjecture. Regarding the mentioned structure
of a partially ordered set we initiate intersection theory of non-commutative
curves. Via the structure of a directed graph we build an analogue to the
classical curve complex used in Teichmueller and Thurston theory. The paper
contains many pictures of graphs and presents an approach to Markov Conjecture
via counting of subgraphs in a graph associated with . Some of the
results proved here were announced in the previous work.Comment: In v4, 65 pages, we have reorganized the paper and removed some
inaccuracies. Sections 2 to 7 are dedicated to general theory and then follow
sections with examples. In the previous version the letter in
the definition of was a set of non-trivial
pairwise non-equivalent triangulated categories. Now we remove the
restriction of non-trivialit
More finite sets coming from non-commutative counting
In our previous papers we introduced categorical invariants, which are,
roughly speaking, sets of triangulated subcategories in a given triangulated
category and their quotients. Here is extended the list of examples, where
these sets are finite. Using results by Geigle, Lenzning, Meltzer, H\"ubner for
weighted projective lines we show that for any two affine acyclic quivers ,
(i.e. quivers of extended Dynkin type) there are only finitely many full
triangulated subctegories in , which are equivalent to
, where is an algebraically closed
field. Some of the numbers counting the elements in these finite sets are
explicitly determined.Comment: 16 pages, In v3 Corollary 5.6 does not depend on any additional
conditions, because in a private communication Professor Helmut Lenzing
confirmed that (21) is correct. The last section 6 and the introduction in
the new version are slightly extended. The reference list is also update
Orlov spectra as a filtered cohomology theory
This paper presents a new approach to the dimension theory and Orlov spectra
of triangulated categories by considering natural filtrations that arise in the
pretriangulated setting.Comment: 27 pages, 2 figure
On K-Stability of Reductive Varieties
G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of
a polarized algebraic variety to the existence of a Kahler metric of constant
scalar curvature. In [Don02] Donaldson partially confirmed it in the case of
projective toric varieties. In this paper we extend Donaldson's results and
computations to a new case, that of reductive varieties.
The changes in the second version are cosmetic
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