Volume 2017, Issue 722

Consider a divisor D with simple normal crossings in a compact Kähler manifold X . It has been known since the work by G. Tian and S. T. Yau that if KX⁢[D]$K_{X}[D]$ is ample there exists on X\D$X\backslash D$ a unique Kähler–Einstein metric with cusp singularities along the divisor (implying completeness and finite volume). We show in this article that a Kähler metric in an arbitrary class with constant scalar curvature and singularities analogous to that constructed by Tian and Yau, is unique in this class when KX⁢[D]$K_{X}[D]$ is ample. This we do by generalizing Chen’s construction of approximate geodesics in the space of Kähler metrics and proving an approximate version of the Calabi–Yau theorem, both independently of the ampleness of KX⁢[D]$K_{X}[D]$.

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler’s multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series [6]. We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein–Kronecker series [9]. Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1;1,1)${(1,1;1,1)}$. Based on such computations, we state two conjectures about MDZV.

Let A be an abelian variety over a number field k and let F be a finite cyclic extension of k of p -power degree for an odd prime p . Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (‘eTNC’) for A , F/k$F/k$ and p as an explicit family of p -adic congruences involving values of derivatives of the Hasse–Weil L -functions of twists of A , normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur and Tate.

We give a proof of openness of versality using coherent functors. As an application, we streamline Artin’s criterion for algebraicity of a stack. We also introduce multi-step obstruction theories, employing them to produce obstruction theories for the stack of coherent sheaves, the Quot functor, and spaces of maps in the presence of non-flatness.

We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K -theory). The technique developed in our study also enables us to compute the K -theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, and determine their K -theory. Then we classify them in terms of the topological full groups, continuous orbit equivalence, strong orbit equivalence, and the crossed products.

We prove some unique factorization results for tensor products of free quantum group factors. They are type III analogues of factorization results for direct products of bi-exact groups established by Ozawa and Popa. In the proof, we first take continuous cores of the tensor products, which satisfy a condition similar to condition (AO), and discuss some factorization properties for the continuous cores. We then deduce factorization properties for the original type III factors. We also prove some unique factorization results for crossed product von Neumann algebras by direct products of bi-exact groups.