Hyperholomorpic connections on coherent sheaves and stability

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for instance, when one deals with higher direct images of holomorphic bundles. We show that such sheaves are stable.Comment: 37 pages, version 11, reference updated, corrected many minor errors and typos found by the refere

Bounded derived categories of very simple manifolds

An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical terms. This is a partial generalization of an impressive result due to Bondal and Van den Bergh.Comment: 11 pages one important references is added, proof of lemma 2.1 (2) and many typos are correcte

Stable bundles on hypercomplex surfaces

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin's and Gualtieri's generalized complex geometry, (4,4)-manifolds are called ``generalized hyperkaehler manifolds''. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.Comment: 17 pages. Version 3.0: reference adde

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure

Potentials for hyper-Kahler metrics with torsion

We prove that locally any hyper-K\"ahler metric with torsion admits an HKT potential.Comment: 9 page

A polynomial bound for untangling geometric planar graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure

Wall-Crossing in Coupled 2d-4d Systems

We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class S, that is, for those theories obtained by compactifying the six-dimensional (0,2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A_1 theories of class S. Finally, we indicate how our results can be used to produce solutions to the A_1 Hitchin equations on the Riemann surface C.Comment: 170 pages, 45 figure

Local and global behaviour of nonlinear equations with natural growth terms

This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results consist of optimal pointwise estimates for positive solutions of such equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto

Nicotine signals through muscle-type and neuronal nicotinic acetylcholine receptors in both human bronchial epithelial cells and airway fibroblasts

BACKGROUND: Non-neuronal cells, including those derived from lung, are reported to express nicotinic acetylcholine receptors (nAChR). We examined nAChR subunit expression in short-term cultures of human airway cells derived from a series of never smokers, ex-smokers, and active smokers. METHODS AND RESULTS: At the mRNA level, human bronchial epithelial (HBE) cells and airway fibroblasts expressed a range of nAChR subunits. In multiple cultures of both cell types, mRNA was detected for subunits that constitute functional muscle-type and neuronal-type pentomeric receptors. Two immortalized cell lines derived from HBE cells also expressed muscle-type and neuronal-type nAChR subunits. Airway fibroblasts expressed mRNA for three muscle-type subunits (α1, δ, and ε) significantly more often than HBE cells. Immunoblotting of HBE cell and airway fibroblast extracts confirmed that mRNA for many nAChR subunits is translated into detectable levels of protein, and evidence of glycosylation of nAChRs was observed. Some minor differences in nAChR expression were found based on smoking status in fibroblasts or HBE cells. Nicotine triggered calcium influx in the immortalized HBE cell line BEAS2B, which was blocked by α-bungarotoxin and to a lesser extent by hexamethonium. Activation of PKC and MAPK p38, but not MAPK p42/44, was observed in BEAS2B cells exposed to nicotine. In contrast, nicotine could activate p42/44 in airway fibroblasts within five minutes of exposure. CONCLUSIONS: These results suggest that muscle-type and neuronal-type nAChRs are functional in airway fibroblasts and HBE cells, that prior tobacco exposure does not appear to be an important variable in nAChR expression, and that distinct signaling pathways are observed in response to nicotine

Motor phenotype of LRRK2 G2019S carriers in early-onset Parkinson disease

Objective: To determine the motor phenotype of LRRK2 G2019S mutation carriers. LRRK2 mutation carriers were previously reported to manifest the tremor dominant motor phenotype, which has been associated with slower motor progression and less cognitive impairment compared with the postural instability and gait difficulty (PIGD) phenotype. Design: Cross-sectional observational study. Setting: Thirteen movement disorders centers. Participants: Nine hundred twenty-five early-onset Parkinson disease cases defined as age at onset younger than 51 years. Main Outcome Measures: LRRK2 mutation status and Parkinson disease motor phenotype: tremor dominant or PIGD. Demographic information, family history of Parkinson disease, and the Unified Parkinson's Disease Rating Scale score were collected on all participants. DNA samples were genotyped for LRRK2 mutations (G2019S, I2020T, R1441C, and Y1699C). Logistic regression was used to examine associations of G2019S mutation status with motor phenotype adjusting for disease duration, Ashkenazi Jewish ancestry, levodopa dose, and family history of Parkinson disease. Results: Thirty-four cases (3.7%) (14 previously reported) were G2019S carriers. No other mutations were found. Carriers were more likely to be Ashkenazi Jewish (55.9% vs 11.9%; P < .001) but did not significantly differ in any other demographic or disease characteristics. Carriers had a lower tremor score (P = .03) and were more likely to have a PIGD phenotype (92.3% vs 58.9%; P = .003). The association of the G2019S mutation with PIGD phenotype remained after controlling for disease duration and Ashkenazi Jewish ancestry (odds ratio, 17.7; P < .001). Conclusion: Early-onset Parkinson disease G2019S LRRK2 carriers are more likely to manifest the PIGD phenotype, which may have implications for disease course