Non-semistable Exceptional Objects in Hereditary Categories

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Bridgeland stability conditions on the acyclic triangular quiver

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Hyperelliptic Szpiro inequality

We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus $g$, the number $N$ of non-separating vanishing cycles and the number $D$ of singular fibers satisfy the inequality $N \leq (4g+2)D$.Comment: LaTeX2e, 27 page

Schematic homotopy types and non-abelian Hodge theory

In this work we use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the \textit{schematization functor} $X \mapsto (X\otimes \mathbb{C})^{sch}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a \textit{Hodge decomposition} on $(X\otimes\mathbb{C})^{sch}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb{C}^{\times \delta}$ on the object $(X\otimes \mathbb{C})^{sch}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group as defined by C.Simpson, and in the simply connected case, the Hodge decomposition on the complexified homotopy groups as defined by J.Morgan and R. Hain. This Hodge decomposition is shown to satisfy a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As a first application we construct a new family of examples of homotopy types which are not realizable as complex projective manifolds. Our second application is a formality theorem for the schematization of a complex projective manifold. Finally, we present conditions on a complex projective manifold $X$ under which the image of the Hurewitz morphism of $\pi_{i}(X) \to H_{i}(X)$ is a sub-Hodge structure.Comment: 57 pages. This new version has been globally reorganized and includes additional results and applications. Minor correction

Density of monodromy actions on non-abelian cohomology

In this paper we study the monodromy action on the first Betti and de Rham non-abelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski dense monodromy orbit. In particular we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.Comment: LaTeX2e, 48 pages, Version substantially revised for publication. A gap in the proof of the density for Lefschetz pencils is fixed. The case of hyperelliptic monodromy is also treated in detai

Some new categorical invariants

In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to CxH for l >= 3. In the present paper we define a new notion of norm, which distinguishes {Db(K(l))}l >= 2. More precisely, to a triangulated category T which has property of a phase gap we attach a non-negative real number epsilon. Natural assumptions on a SOD T=T1,T2 imply epsilon}. Using the norm we define a topology on the set of equivalence classes of proper triangulated categories with a phase gap, such that the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures that [Db(point)]is an element of Cl([Db(S)]). Categories in a neighborhood of Db(K(l)) have the property that Db(K(l)) is embedded in each of them. We view such embeddings as non-commutative curves in the ambient category and introduce categorical invariants based on counting them. Examples show that the idea of non-commutative curve-counting opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi-Yau curve-counting, where the entities we count are a Calabi-Yau modification of Db(K(l)). In the end we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory