In this paper, we study the Pontryagin numbers of 24 dimensional String manifolds. In partic- ular, we find representatives of an integral basis of the String cobrodism group at dimension 24, based on the work of Mahowald and Hopkins (The structure of 24 dimensional manifolds having normal bundles which lift to B O [8], from “Recent progress in homotopy theory” (D. M. Davis, J. Morava, G. Nishida, W. S. Wilson, N. Yagita, editors), Contemp. Math. 293, Amer. Math. Soc., Providence, RI, 89-110, 2002), Borel and Hirzebruch (Am J Math 80: 459–538, 1958) and Wall (Ann Math 75:163–198, 1962). This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the 2-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins and Singer (J Differ Geom 70:329–452, 2005), and also the 3- primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner.

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.

Inthisarticle,weprovideasufficientandnecessaryconditiononparabolic isometries of positive translation lengths on complete visibility CAT(0) spaces. One of the consequences is that each parabolic isometry of a complete simply connected visibility manifold of nonpositive sectional curvature has zero translation length. Applications on the geometry of open negatively curved manifolds will also be discussed.

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1 structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.

We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.