Volume 2013, Issue 675

The goal of the present paper is to show the transformation formula of Donaldson–Thomas invariants on smooth projective Calabi–Yau 3-folds under birational transformations via categorical method. We also generalize the non-commutative Donaldson–Thomas invariants, introduced by B. Szendrői in a local (−1, −1)-curve example, to an arbitrary flopping contraction from a smooth projective Calabi–Yau 3-fold. The transformation formula between such invariants and the usual Donaldson–Thomas invariants are also established. These formulas will be deduced from the wall-crossing formula in the space of weak stability conditions on the derived category.

We construct an equivalence of categories from a strong categorical sl (2) action, following the work of Chuang–Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary Grassmannians.

We define n -angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller's parametrization of pre-triangulations extends to pre- n -angulations. We obtain a large class of examples of n -angulated categories by considering ( n  − 2)-cluster tilting subcategories of triangulated categories which are stable under the ( n  − 2)-nd power of the suspension functor. As an application, we show how n -angulated Calabi–Yau categories yield triangulated Calabi–Yau categories of higher Calabi–Yau dimension. Finally, we sketch a link to algebraic geometry and string theory.

We give a conjectural description for the cone of effective divisors of the Grothendieck–Knudsen moduli space M ̅ 0, n of stable rational curves with n marked points. Namely, we introduce new combinatorial structures called hypertrees and show that they give exceptional divisors on M ̅ 0, n with many remarkable properties.

In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension n  ≧ 3 is locally a warped product with ( n  − 1)-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.

We prove that for a wide class of motives the local and global non-commutative Tamagawa number conjectures of Fukaya and Kato imply certain explicit restrictions on the Galois structure of Selmer modules (in the sense of Bloch and Kato). We thereby obtain a variety of new, concrete and very general conjectures concerning the structures of Selmer modules. In several important cases we then prove these conjectures.

In previous work, we proved a result on the equivariant Fitting ideal of the ℓ-adic realization of the Picard 1-motive attached to an abelian covering of curves defined over a finite field. In this paper, we build upon this work to deduce results on the equivariant Fitting ideal of the Tate modules of the Jacobian of the top curve, and on the equivariant Fitting ideal of the (dual of the) degree zero class group of the top curve. At the end, we discuss refinements of Brumer's conjecture and consider examples.