Related papers: Projective embedding of pairs and logarithmic K-st…

Let $\sigma_i$, $i=1,\ldots,n$, denote positive Borel measures on $\mathbb{R}^d$, let $\mathcal{D}$ denote the usual collection of dyadic cubes in $\mathbb{R}^d$ and let $K:\,\mathcal{D}\to[0,\infty)$ be a~map. In this paper we give…

We begin the study of Khovanov-Lauda-Rouquier type algebras associated to moduli stacks of coherent sheaves on smooth projective curves. We consider the case of $\mathbb{P}^1$ and define, for any pair $(r,d)$ of a rank and a degree, the KLR…

Supplementing and expanding classical results, for compact spaces $K$ and $L$, $L$ metric, and their Banach spaces $\mathcal{C}(L)$ and $\mathcal{C}(K)$ of continuous real-valued functions, we provide several characterizations of the…

Let $X$ be a K\"ahler manifold with semi-ample canonical bundle $K_X$. It is proved by Jian-Shi-Song that for any K\"ahler class $\gamma$, there exists $\delta>0$ such that for all $t\in (0, \delta)$ there exists a unique cscK metric $g_t$…

In this paper, we prove that if a compact K\"ahler manifold $X$ has a smooth Hermitian metric $\omega$ such that $(T_X,\omega)$ is uniformly RC-positive, then $X$ is projective and rationally connected. Conversely, we show that, if a…

In this paper, we study the cohomology of vector bundles on projective space defined as kernels or cokernels of general maps $V_1 \to V_2$, where the $V_i$ are direct sums of line bundles or certain exceptional bundles. We prove an…

Given an automorphic line bundle ${\mathcal O}_X(k)$ of weight $k$ on the Drinfel'd upper half plane $X$ over a local field $K$, we construct a ${\rm GL}_2(K)$-equivariant integral lattice ${\mathcal O}_{\widehat{\mathfrak X}}(k)$ in…

We generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces `tout court'. For a class of such spaces, we explicitly construct families of Fredholm…

Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power…

Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact K\"ahler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-\varphi_F}$. Via the use of the…

In this paper we consider the K-theory of smooth algebraic stacks, establish lambda and gamma operations, and show that the higher K-theory of such stacks is always a pre-lambda-ring, and is a lambda-ring if every coherent sheaf is the…

Let $(X,\Delta)$ be a smooth complex projective simple normal crossing pair of dimension $n\geq 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+\Delta$ is not nef, then precisely one of the following…

Let $C$ be a smooth irreducible projective curve and let $(L,H^0(C,L))$ be a complete and generated linear series on $C$. Denote by $M_L$ the kernel of the evaluation map $H^0(C,L)\otimes\mathcal O_C\to L$. The exact sequence $0\to M_L\to…

We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for nontrivial weights) is isomorphic to the ring of integral…

Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$. For each $d <\eta(X)$, where $\eta(X)$ is the gonality of $X$, the symmetric product $\text{Sym}^d(X)$ embeds into $\text{Pic}^d(X)$ by sending an effective…

Let $L$ be a holomorphic line bundle on a compact complex manifold $X$ of dimension $n,$ and let $e^{-\phi}$ be a continuous metric on $L.$ Fixing a measure $d\mu$ on $X$ gives a sequence of Hilbert spaces consisting of holomorphic sections…

We show that the CM line bundle on a proper family parametrizing specially K-stable varieties with maximal variation is ample. As an application, we show projectivity of any proper subspace of the coarse moduli space of uniformly…

For a polarized K\"ahler manifold $(X, L)$, we show the equivalence between relative balanced embeddings introduced by Mabuchi and $\sigma$-balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT…

We consider a compact CR manifold with a transversal CR locally free circle action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szeg\H{o} kernel of the CR sections in the high tensor powers admits a…

Let $M$ be a Carath\'eodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete K\"ahler, we show that $M$ admits the Bergman metric. When $M$ is…