Let G=expg be an exponential solvable Lie group with Lie algebra g, let H=exph be an analytic subgroup of G with Lie algebra h, and let χ be a unitary character of H. As a continuation of our study in the nilpotent case, we investigate some problems around a monomial representation τ=indHGχ of G, where χ(expX)=χf(expX)=eif(X)(X∈h) for some f∈g∗. Supposing that τ is of discrete type, our object is the algebra Dτ(G∕H) of G-invariant differential operators on the line bundle associated to the data (G,H,χ). As in the nilpotent setting, we show that the algebra Dτ(G∕H) is commutative or even more is isomorphic to the algebra C[Γτ]H of H-invariant polynomial functions on Γτ=f+h⊥.
We prove that the Generalized Mukai Conjecture holds for Fano manifolds X such that dim X = 3iX, where iX is the pseudoindex of X, provided that either any dominating family of rational curves on X is not unsplit and X is rationally connected by two "special" families arising in a series of rational fibrations, or X has an extremal ray associated with a contraction of fiber type. Moreover, we give partial results if X has an extremal ray of length greater than or equal to iX + 1 associated with a divisorial contraction.
In this short note, we show that there is a smooth projective surface which dominates infinitely many isomorphism classes of normal projective hyperbolic surfaces of general type. This shows the limitations in extending Severi's classical finiteness theorem for curves to higher-dimensional varieties. Our proof is constructive and relies on the birational geometry of surfaces.
The close relation between 2-term Gorenstein silting complexes in the bounded Gorenstein derived categories and 2-term silting complexes in the bounded derived categories is first showed. Then, as the module-theoretic counterpart of 2-term Gorenstein silting complexes for finite dimensional algebras of finite CM-type, the relative rigid modules are introduced and investigated. Furthermore, the relations among 2-term Gorenstein silting complexes, t-structures, and torsion pairs are exhibited. The counterpart of Brenner-Butler theorem and the global dimension of the endomorphism algebra of a 2-term Gorenstein silting complex are described.
In this paper, we completely classify 3-dimensional complete lambda-translators with constant squared norm S of the second fundamental form and constant f3 in the Euclidean space IIg4 and the Minkowski space IIg41, where hijare components of the second fundamental form, S = & sum;i,jh2ijand f3 = & sum;i,j,k hijhjkhki.
