We construct gradient K\"ahler-Ricci solitons on Ricci-flat K\"ahler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the results generalize constructions of Cao and Feldman-Ilmanen-Knopf.

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer--Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich--Soibelman, Gross--Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer--Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer--Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.

In this paper we study the analytic torsion and the L2-torsion of compact locally symmetric manifolds. We consider the analytic torsion with respect to representations of the fundamental group which are obtained by restriction of irreducible representations of the group of isometries of the underlying symmetric space. The main purpose is to study the asymptotic behavior of the analytic torsion with respect to sequences of representations associated to rays of highest weights.

We construct balanced metrics on the family of non-Khler Calabi-Yau threefolds that are obtained by smoothing after contracting ( 1, 1) -rational curves on a Khler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to the connected sum of ( 1, 1) copies of ( 1, 1) .

We prove the factoriality of a nodal hypersurface in P4 of degree d that has at most 2(d ¡ 1)2=3 singular points, and we prove the factoriality of a double cover of P3 branched over a nodal surface of degree 2r having less than (2r ¡ 1)r singular points.