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 $D^b(K(l))$ as a non-commutative curve, where $K(l)$ is the Kronecker quiver with $l$-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 $D^b(K(l))$. 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 $D^b(Q)$ for two affine quivers, for $A_n$ and $D_4$ we derive formulas for counting of the subcategories of type $D^b(A_k)$ in $D^b(A_n)$, whereas for the two affine quivers and for $D_4$ we determine and count all generated by an exceptional collection subcategories. Estimating the numbers counting non-commutative curves in $D^b({\mathbb P}^2)$ 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 $D^b(P^2)$. 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 $\mathcal J$ in the definition of $C_{\mathcal J, P}(\mathcal T)$ was a set of non-trivial pairwise non-equivalent triangulated categories. Now we remove the restriction of non-trivialit

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

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 $Q$, $Q'$ (i.e. quivers of extended Dynkin type) there are only finitely many full triangulated subctegories in $D^b(Rep_{\mathbb K}(Q))$, which are equivalent to $D^b(Rep_{\mathbb K}(Q'))$, where ${\mathbb K}$ 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

Complex projective surfaces and infinite groups

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be non-residually finite. Using the construction we also suggest a series of potential counterexamples to the Shafarevich conjecture which claims that the universal covering of smooth projective variety is holomorphically convex. The examples are only potential since they depend on group theoretic questions, which we formulate, but we do not know how to answer. At the end we formulate an arithmetic version of the Shafarevich conjecture.Comment: 29 pages, some comments and examples added LaTeX 2.0