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 introduce a new equivalence relation for complete alge- braic varieties with canonical singularities, generated by birational equivalence, by flat algebraic deformations (of varieties with canon- ical singularities), and by quasi-´etale morphisms, i.e., morphisms which are unramified in codimension 1. We denote the above equivalence by A.Q.E.D. : = Algebraic-Quasi-´ Etale- Deformation. A completely similar equivalence relation, denoted by C-Q.E.D., can be considered for compact complex spaces with canonical sin- gularities. By a recent theorem of Siu, dimension and Kodaira dimension are invariants for A.Q.E.D. of complex varieties. We address the interesting question whether conversely two al- gebraic varieties of the same dimension and with the same Ko- daira dimension are Q.E.D. - equivalent (A.Q.E.D., or at least C-Q.E.D.), the answer being positive for curves by well known results. Using Enriques’ (resp. Kodaira’s) classification we show first that the answer to the C-Q.E.D. question is positive for special algebraic surfaces (those with Kodaira dimension at most 1), resp. for compact complex surfaces with Kodaira dimension 0, 1 and even first Betti number. The appendix by S¨onke Rollenske shows that the hypothesis of even first Betti number is necessary: he proves that any sur- face which is C-Q.E.D.-equivalent to a Kodaira surface is itself a Kodaira surface. We show also that the answer to the A.Q.E.D. question is pos- itive for complex algebraic surfaces of Kodaira dimension ≤ 1. The answer to the Q.E.D. question is instead negative for sur- faces of general type: the other appendix, due to Fritz Grunewald, is devoted to showing that the (rigid) Kuga-Shavel type surfaces of general type obtained as quotients of the bidisk via discrete groups constructed from quaternion algebras belong to countably many distinct Q.E.D. equivalence classes.

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We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold$X$. Given a big (1, 1)-cohomology class$α$on$X$(i.e. a class that can be represented by a strictly positive current) and a positive measure$μ$on$X$of total mass equal to the volume of$α$and putting no mass on pluripolar sets, we show that$μ$can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in$α$. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if$μ$has$L$^{1+$ε$}-density with respect to Lebesgue measure. If$μ$is smooth and positive everywhere, we prove that$T$is smooth on the ample locus of$α$provided$α$is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.

The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.