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Register a new userTorus fibrations and localization of index III
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This paper is the third of the series concerning the localization of the index of Dirac-type operators. In our previous papers we gave a formulation of index of Dirac-type operators on open manifolds under some geometric setting, whose typical example was given by the structure of a torus fiber bundle on the ends of the open manifolds. We introduce two equivaria
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We give a framework of localization for the index of a Dirac-type operator on an open manifold. Suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets, each of which has a structure of the total space of a torus bundle. Under an acyclic condition we define the index of the Dirac-type operator by using the Witten-type deformation, and show that the index has several properties, such as excision property and a product formula. In particular, we show that the index is localized on the compact set.
We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Wittens deformation applied to a Hilbert bundle.
We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.
We examine the action of the fundamental group $Gamma$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $widetilde{mathcal B}$ on $m$ strings on a suitable space of $mtimes m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${mathcal S}_c:widetilde{mathcal B}to GL_m({mathbb{Z}}[Gamma])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information.
We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, SKT and astheno-Kahler metrics. We prove that the twistor spaces of compact hyperkahler and negative quaternionic-Kahler manifolds do not admit astheno-Kahler metrics. Then we provide examples of astheno-Kahler structures on toric bundles over Kahler manifolds. In particular, we find examples of compact complex non-Kahler manifolds which admit a balanced and an astheno-Kahler metrics, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups $SU(3)$ and $G_2$ admit SKT and astheno-Kahler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space $M$ with invariant volume admits a balanced metric, then its first Chern class $c_1(M)$ does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.
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