In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle 2πβ along the divisor, then for any β ′ sufficiently close to β, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle 2πβ ′ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence in [36,37]. As corollaries, we give parabolic proofs of Donaldson's openness theorem [17] and his existence conjecture [18] for the conical Kähler-Einstein metrics with positive Ricci curvatures. IntroductionSince the conical Kähler-Einstein metrics play an important role in the solution of the Yau-Tian-Donaldson's conjecture which has been proved by Chen-Donaldson-Sun [6][7][8] and Tian [51], the existence and geometry of the conical Kähler-Einstein metrics have been widely concerned. The conical Kähler-Einstein metrics have been studied by Berman [1], Brendle [3], Campana-Guenancia-Pȃun [4], Donaldson [17], Guenancia-Pȃun [21], Guo-Song [22,23], Jeffres [25], Jeffres-Mazzeo-Rubinstein [26], Li-Sun [32], Mazzeo [38], Song-Wang [48], Tian-Wang [52] and Yao [58] etc. For more details, readers can refer to Rubinstein's article [44].The conical Kähler-Ricci flows were introduced to attack the existence of the conical Kähler-Einstein metrics. These flows were first proposed in Jeffres-Mazzeo-Rubinstein's paper (see Section 2.5 in [26]), then Song-Wang (conjecture 5.2 in [48]) made a conjecture on the relation between the convergence of these flows and the greatest Ricci lower bounds of the manifolds. Then the existence, regularity and limit behavior of the conical Kähler-Ricci flows have been studied by Chen-Wang [9,10], Edwards [19,20], Liu-Zhang [34], Liu-Zhang [36,37], Nomura [39], Shen [45, 46], Wang [57] and Zhang [62, 63] etc.Let M be a Fano manifold with complex dimension n, ω 0 ∈ c 1 (M ) be a smooth Kähler metric and D ∈ | − λK M | (0 < λ ∈ Q) be a smooth divisor.
In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, under some assumptions on the divisor and α-invariant, we get the properness of the modified log K-energy and the existence of conical Kähler-Ricci soliton with suitable cone angle.1 XISHEN JIN, JIAWEI LIU, AND XI ZHANG is a smooth Kähler metric invariant under the action of Φ Im X , where Φ Im X is the one-parameter transformations subgroup of Aut(M ) generating by Im X, i.e.> 0 in the sense of current}. As in [5], we define the following function subspace of H(M, ω 0 ):We define K 0 X (ω 0 ) to be the space of smooth semipositive (1, 1)-forms cohomology to ω 0 , i.e.is a subspace of Kähler metrics defined as follow:Definition 1.1. We define the following invariant with respect to X,Note that ω is a closed form and X is holomorphic, we have that ∂(i X ω) = 0. According to the Hodge decomposition theorem and the property of Fano manifold, we can find a smooth real-valued function θ X (ω) such that for ω ∈ K X (ω 0 ),and θ X (ω) satisfies the normalization M e θX (ω) ω n = M ω n 0 .We will take notation that θ X = θ X (ω 0 ) in the whole paper without special instruction. By direct computation, we get that θ X (ω ϕ ) = θ X + X(ϕ).Definition 1.2. For any (1, 1)-form η ∈ (1 − β)K 0 X (ω 0 ), we say a Kähler metric ω ∈ K X (ω 0 ) is a twisted Kähler-Ricci soliton with respect to η if it satisfiesRemark 1.2. It is easy to see that finding the twisted Kähler-Ricci soliton as (1.4) is equivalent to solving the following Monge-Ampère equation:where h ω is the Ricci potential defined by (1.6) Ric(ω) − βω − η = √ −1∂∂h ω normalized such that M e hω ω n = M ω n . And we just consider the case when β is nonnegative, since the equation (1.5) is solvable according to the celebrated work of Aubin [1] and Yau [31] on the other case.
Background The CRISPR/Cas12a and CRISPR/Cas13d systems are widely used for fundamental research and hold great potential for future clinical applications. However, the short half-life of guide RNAs (gRNAs), particularly free gRNAs without Cas nuclease binding, limits their editing efficiency and durability. Results Here, we engineer circular free gRNAs (cgRNAs) to increase their stability, and thus availability for Cas12a and Cas13d processing and loading, to boost editing. cgRNAs increases the efficiency of Cas12a-based transcription activators and genomic DNA cleavage by approximately 2.1- to 40.2-fold for single gene editing and 1.7- to 2.1-fold for multiplexed gene editing than their linear counterparts, without compromising specificity, across multiple sites and cell lines. Similarly, the RNA interference efficiency of Cas13d is increased by around 1.8-fold. In in vivo mouse liver, cgRNAs are more potent in activating gene expression and cleaving genomic DNA. Conclusions CgRNAs enable more efficient programmable DNA and RNA editing for Cas12a and Cas13d with broad applicability for fundamental research and gene therapy.
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