D. H. Phong, Jacob Sturm
In this paper, we the improve the bound for the moment map derivative proved by Donaldson in his recent proof of the Hilbert-Mumford stability of complex manifolds with constant scalar curvature. The proof depends on the identification of Donaldson's symplectic form with the curvature of a certain Deligne pairing.
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Let $X \rightarrow S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over complex numbers. Let L_0, L_1, .... L_{n-1}, L_{n} be line bundles over $X$. There is a natural isomorphism of the Deligne pairing $<L_{0},...,L_{n}>$ with the determinant line bundle ${\rm Det}(\otimes_{i=0}^{n} (L_i- {\mathcal O}_{X}))$.
Let $X\to B$ be a proper flat morphism between smooth quasi-projective varieties of relative dimension $n$, and $L\to X$ a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (π_* L^k)$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K$. This generalizes a theorem of Deligne which holds for families of relative dimension one. As a corollary, we show that when $X$ is smooth (or, more generally, if $X$ fits in a smooth family), the line bundle associated to $X\to B$, which was introduced by the …
We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to b…
Let $X\rightarrow S$ be a smooth projective surjective morphism of relative dimension $n$, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L\rightarrow X$ be a relatively very ample line bundle. For every sufficiently large positive integer $m$, there is a canonical isomorphism of the Deligne pairing $\langle L ,\cdots , L\rangle\rightarrow S$ with the determinant line bundle ${\rm Det}((L- {\mathcal O}_{X})^{\otimes (n+1)}\otimes L^{\otimes m})$ \cite{PRS}. If we fix a hermitian structure on $L$ and a relative Kähler form on $X$, then each of the line bundles ${\rm Det}((L- {\ma…
For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyper…
Let V be a finite dimensional complex vector space and V^* its dual and let X in P(V) be a smooth projective variety of dimension n and degree d at least two. For a generic n-tuple of hyperplanes H_1,...,H_n in P(V^*)^n, the intersection of X with H_1,...,H_n consists of d distinct points. We define the "discriminant of X", to be the the set D(X) of n-tuples for which the set-theoretic intersection is not equal to d points. Then D(X) is a hyper surface in P(V^*)^n and the set of defining polynomials, which is a one-dimensional vector space, is called the "discriminant line". We show that this …
Let h^{*} be a multiplicative cohomology theory, h_{*} its dual homology theory and \hat{h}^{*} a differential refinement. We first construct the natural pairing between h_{*} and the flat part of \hat{h}^{*}, generalizing the holonomy of a flat Deligne cohomology class. Then, in order to generalize the holonomy of any Deligne cohomology class, we define the generalized Cheeger-Simons characters. The latter are functions from suitably defined differential cycles to the cohomology ring of the point, such that the value on a trivial cycle only depends on the curvature.
We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$--gerbe $Λ_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of `linear equivalence to zero'. In this setting, Abel's theorem remains valid. Moreover we generalize the classical Inversion Theorem for the Abel--Jacobi map, thereby proving th…
Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models
本文修正了arXiv:1903.09181中命题13的证明错误，并证明了当4维流形M的边界拓扑嵌入且局部平坦时，其第一同调群的torsion部分为直接双重，扩展了Hantzsche定理。
Covering complexity, scalar curvature, and quantitative $K$-theory
本文建立了Riemannian spin流形的覆盖复杂度与其标量曲率正下界之间的关系，该方法利用了定量算子$K$-理论与Lipschitz拓扑$K$-理论的配对及先前的消没定理。
Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the ind…
Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature
本文为李群等距作用的流形建立等变椭圆算子及 Callias 型算子指标理论，并由此给出非余紧流形上正标量曲率度量的存在性障碍。
Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature
本文将 Higson–Roe 解析手术序列映射到非交换 de Rham 同调长正合序列，构造高阶 rho 数及相关配对，并研究微分同胚作用，从而得到 spin 流形正标量曲率度量模空间的新结论。
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