In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space M of smoothable Kähler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on M and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kähler-Einstein Fano mani...

BACKGROUND Elimination of hospital-acquired infections is an important patient safety goal. SETTING All 174 medical, cardiac, surgical and mixed Veterans Administration (VA) intensive care units (ICUs). INTERVENTION A centralised infrastructure (Inpatient Evaluation Center (IPEC)) supported the practice bundle implementation (handwashing, maximal barriers, chlorhexidinegluconate site disinf...

X(L) = {x ∈ X(L) such that Gx is finite and G · x is closed in X (L)} , where Gx is the stabilizer of x. It turns out that π −1 (π(x)) = G · x, for all x ∈ X(L). We refer to [7] or [8] for the classical properties of this quotient. Observe that this GIT-quotient is not canonical: it depends on a choice of an ample G-linearized line bundle L over X (sometimes called a polarization of X). During ...

Let S be a surface over C with only normal crossing singularities, abbreviated as nc. That is, each point of S is analytically isomorphic to one of 3 local models: smooth point (x = 0) ⊂ C, double nc (xy = 0) ⊂ C or triple nc (xyz = 0) ⊂ C. The normalization n : S̄ → S is smooth and the preimage of the singular locus D ⊂ S is a nc curve D̄ ⊂ S̄. The dualizing sheaf (or canonical line bundle) ωS is...

Let L be a very ample line bundle on a smooth curve C of genus g with 3g+3 2 < degL ≤ 2g − 5. Then L is normally generated if degL > max{2g+2−4h(C,L), 2g− g−1 6 −2h(C,L)}. Let C be a triple covering of genus p curve C′ with C φ → C′ and D a divisor on C′ with 4p < degD < g−1 6 − 2p. Then KC(−φ ∗D) becomes a very ample line bundle which is normally generated. As an application, we characterize s...

Let π : X → B be a holomorphic submersion between compact Kähler manifolds of any dimensions, whose fibres and base have no non-zero holomorphic vector fields and whose fibres admit constant scalar curvature Kähler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature Kähler metric on X. The condition involves the CM-line bundle—a certai...

Let X be a semi-stable arithmetic surface over the spectrum S of the ring of integers in a number field K. We assume that the generic fiber XK is geometrically irreducible and has positive genus. Consider a line bundle L on X , which is non-negative on every vertical fiber and equipped with an admissible metric (in the sense of Arakelov [A]) of positive curvature. In [S] we have shown that any ...

The purpose of this article is to study the minimal free resolution of homogeneous coordinate rings of elliptic ruled surfaces. Let X be an irreducible projective variety and L a very ample line bundle on X , whose complete linear series defines the morphism φL : X −→ P(H (L)) Let S = ⊕∞ m=0 S H(X,L) and R(L) ⊕∞ m=0 H (X,L). Since R(L) is a finitely generated graded module over S, it has a mini...

In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal IA of the embedding of A via L⊗n are all quadratic; a possible choice...

Explicit combinatorial cancellation-free rules are given for the product of an equivariant line bundle class with a Schubert class in the torus-equivariant K-theory of a KacMoody flag manifold. The weight of the line bundle may be dominant or antidominant, and the coefficients may be described either by Lakshmibai-Seshadri paths or by the λ-chain model of the first author and Postnikov [LP, LP1...