Asymptotically cylindrical Ricci-flat manifolds

Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindrical manifold with Ricci-flat metric can have at most two cylindrical ends. In the case where there are two such cylindrical ends, then there is reduction in the holonomy group Hol and is a cylinder.

     

Deformations and derived equivalences

Suppose and are block algebras of finite groups over a complete local commutative Noetherian ring whose residue field is a field of positive characteristic. We prove that a split-endomorphism two-sided tilting complex (as introduced by Rickard) for the derived categories of bounded complexes of finitely generated modules over , resp. , preserves the versal deformation rings of bounded complexes of finitely generated modules over , resp. .

     

Remarks on defective Fano manifolds

This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016).     

Remarks on Einstein-like Hermitian manifolds

We investigate some relations among Einstein-like conditions on Hermitian manifolds.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

Remarks on subharmonic functions on riemannian manifolds

Sunto Si prova un teorema di limitatezza per funzioni subarmoniche su una varietà riemanniana completa M e soddisfacenti la condisione (A) (cfr. anche [8])e se ne danno alcune applicazioni allo studio del volume di M, dello spazio delle forme armoniche a coefficienti in fibrati vettoriali su M, di cette mappe armoniche definite su M.     

Double Hall algebras and derived equivalences

We show that the reduced Drinfeld double of the Ringel-Hall algebra of a hereditary category is invariant under derived equivalences. By associating an explicit isomorphism to a given derived equivalence, we also extend the results of Burban and Schiffmann [5] and [006], Sevenhant and Van den Bergh (1999) [25], and Xiao and Yang (2001) [31].     

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

Relative derived equivalences and relative homological dimensions

Let A be a small abelian category. For a closed subbifunctor F of Ext_A¹(-,-), Buan has generalized the construction of Verdier's quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.     

Constructions of derived equivalences for algebras and rings

In this article, we shall survey some aspects of our recent (or related) constructions of derived equivalences for algebras and rings.     

Remarks on BCOV invariants and degenerations of Calabi-Yau manifolds

For a one parameter family of Calabi-Yau threefolds, Green et al.(2009) expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper,we show that the total singularities can be expressed by the sum of asymptotic values of BCOV(BershadskyCecotti-Ooguri-Vafa) invariants, studied by Fang et al.(2008). On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.     

Dominant dimensions,derived equivalences and tilting modules

We show that for every ? > 0 there exist δ > 0 and n₀ ∈ ? such that every 3-uniform hypergraph on n ≥ n₀ vertices with the property that every k-vertex subset, where k ≥ δn, induces at least \(\left( {\frac{1}{2} + \varepsilon } \right)\left( {\begin{array}{*{20}c} k \\ 3 \\ \end{array} } \right)\) edges, contains K₄^? as a subgraph, where K₄^? is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erd?s and Sós. The constant 1/4 is the best possible.     

Higher-dimensional orders,graph-orders,and derived equivalences

Green-orders (tree-orders) in the classical one-dimensional case are the setting, to understand p-adic blocks with cyclic defect of finite groups. Blocks with “cyclic defect” of Hecke orders however, are Green-orders over two-dimensional rings. Hecke orders of dihedral groups of order divisible by 4 are even defined over a three-dimensional ring. We extend the notion of Green-orders to orders associated to a locally embedded graph instead of a tree, and to general complete regular local noetherian ground rings of finite dimension. We extend the result, that classical tree-orders are derived equivalent to star-orders. We then use these results to clarify the derived equivalence classes of tame algebras of Dihedral type.