Volume 2021, Issue 771

In this paper, we construct virtual cycles on moduli spaces of solutions to the perturbed gauged Witten equation over a fixed smooth r -spin curve, under the framework of [G. Tian and G. Xu, Analysis of gauged Witten equation, J. reine angew. Math. 740 (2018), 187–274]. Together with the wall-crossing formula proved in the companion paper [G. Tian and G. Xu, A wall-crossing formula for the correlation function of gauged linear σ-model, preprint], this paper completes the construction of the correlation function for the gauged linear σ-model announced in [G. Tian and G. Xu, Correlation functions in gauged linear σ-model, Sci. China Math. 59 (2016), 823–838] as well as the proof of its invariance.

We study the behaviour of the Arakelov metric on a smooth curve under semistable degeneration. The final result is a complicated formula involving the local discriminants of the singularities, and the graph governing the degeneration.

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time t → 0 - {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as t → - ∞ {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions S J × ℝ n - J {S^{J}\times\mathbb{R}^{n-J}} , 1 ≤ J ≤ n - 1 {1\leq J\leq n-1} . These results are the analog of the corresponding results in Ricci flow ( J = n - 1 {J=n-1} ) and mean curvature flow.

In this paper we study the structure of the pointed-Gromov–Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger–Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any non-collapsing Ricci shrinker limit space is strongly convex, inspired by Colding–Naber’s original idea of parabolic smoothing of the distance functions.

Let X be a compact Kähler manifold. Given a big cohomology class { θ } {\{\theta\}} , there is a natural equivalence relation on the space of θ-psh functions giving rise to 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} , the space of singularity types of potentials. We introduce a natural pseudo-metric d 𝒮 {d_{\mathcal{S}}} on 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the d 𝒮 {d_{\mathcal{S}}} -topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a ℤ {{\mathbb{Z}}} -cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

Let f : ℂ → X {f:\mathbb{C}\to X} be a transcendental holomorphic curve into a complex projective manifold X . Let L be a very ample line bundle on X . {X.} Let s be a very generic holomorphic section of L and D the zero divisor given by s . {s.} We prove that the geometric defect of D (defect of truncation 1) with respect to f is zero. We also prove that f almost misses general enough analytic subsets on X of codimension 2.

We construct smooth rational real algebraic varieties of every dimension ≥ 4 {\geq 4} which admit infinitely many pairwise non-isomorphic real forms.