For stratified Mukai flops of type An,k,D2k+1 and E6,I , it is shown that the fiber product induces isomorphisms on Chow motives. In contrast to (standard) Mukai flops, the cup product is generally not preserved. For An,2, D5 and E6,I flops, quantum corrections are found through degeneration/deformation to ordinary flops.

H. Hopf showed that the only constant mean curvature sphere S2 immersed in R3 is the round sphere. The K¨ahler framework is an adequate approach to generalize Hopf’ s theorem to higher dimensions. When ϕ : M → Rn is an isometric immersion from a K¨ahler manifold, the complexified second fundamental form α splits according to types. The (1, 1) part of the second fundamental form plays the role of the mean curvature for surfaces and will be called the pluri-mean curvature pmc. Therefore isometric immersions with parallel pluri-mean curvature (ppmc isometric immersions) generalize in a natural way the cmc immersions. It is a standard fact that R8 is the smallest space where CP2 can be embedded. The aim of this work is to generalize Hopf’s theorem proving in particular that the only ppmc isometric immersion from CP2 into R8 is the standard immersion.

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For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture [33, 38, 3] and its refinement [34], we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with b+ = 1 in terms of modular forms. This formula was proved earlier in [19] more generally for simply connected 4-manifolds with b+ = 1, as- suming the Kotschick-Morgan conjecture, and it was also derived by physical arguments in [31].

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into IRn, n = 3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound Wn,p depending on codimension and genus p of the Riemann surface. For tori in codimension 1, we know W3,1 = 8 .