We give a generalizations of lower Ricci curvature bound in the framework of graphs. We prove that the Ricci curvature in the sense of Bakry and Emery is bounded below by 1 on locally finite graphs. The Ricci flat graph in the sense of Chung and Yau is proved to be a graph with Ricci curvature bounded below by zero. We also get an estimate for the eigenvalue of Laplace operator on finite graphs: 1 dD (exp (dD+ 1) 1)

We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of An-singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in [30] and the homological mirror symmetry for An-singularities.

We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a point in Teichmüller space can recover the marking and geometry of this marked surface. We then translate the results concerning the infinitesimal structures to global geometric statements for the Thurston metric, most notably deriving rigidity statements for the Thurston metric analogous to the celebrated Royden theorem.

Let (X, g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X, J) is equal to −∞ and the canonical bundle KX is not pseudo-effective. We also introduce the complex Yamabe number λc(X) for compact complex manifold X, and show that if λ_c(X) is greater than 0, then κ(X) is equal to −∞; moreover, if X is also spin, then the Hirzebruch A-hat genus \hat{A}(X) is zero.

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