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Sean Timothy Paul

Publications and source records attributed to Sean Timothy Paul.

14 recordsLinked to original sources
arXiv2021

Mahler measures, Stable Pairs, and the Global coercive estimate for the Mabuchi Functional

We show that the Mabuchi energy of any polarized manifold (X,L) is (bounded below) proper on the full space of Kahler metrics in the first Chern class of L if and only if (X,L) is asymptotically (semi)stable. In particular it now follows from work of Xiuxiong Chen and Jinguri Cheng that there exists a cscK metric in the first Chern class of L if and only if (X,L) is asymptotically stable, provided the reduced automorphism group of (X,L) is finite.

math.DG↗
arXiv2012

CM Stability of Projective varieties

We develop the connection between equivariant completions of algebraic homogeneous spaces of reductive groups and lower bounds for the Mabuchi energy of a polarized manifold over the space of Bergman metrics. We provide a new definition of Tian's CM Polarization and discuss its properties.

math.AG↗
arXiv2010

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori H of G .

math.DG↗
arXiv2008

On a result of Gelfand, Kapranov, and Zelevinsky

In this paper I give new elementary proofs of basic results of Gelfand, Kapranov and Zelevinskywhich express discriminants and resultants in terms of determinants of direct images of Cayley-Koszul complexes of sheaves.

math.AG↗
arXiv2008

Projective duality and K-energy asymptotics

Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plane curves, this energy functional reduces to the standard action functionals of Kahler geometry.

math.DG↗
arXiv2008

CM Stability And The Generalized Futaki Invariant II

The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family $\mathbf{X}\overset{p}{\to} S$. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map $p$. Properness of the K-energy implies that the generalized Futaki invariant is strictly negative.

math.DG↗
arXiv2007

Higher Energies in Kahler Geometry I

Let $X\hookrightarrow \cpn $ be a smooth complex projective variety of dimension $n$. Let $λ$ be an algebraic one parameter subgroup of $G:=\gc$. Let $ 0\leq l\leq n+1$. We associate to the coefficients $F_{l}(λ)$ of the normalized weight of $λ$ on the $mth$ Hilbert point of $X$ new energies $F_{\om,l}(\vp)$. The (logarithmic) asymptotics of $F_{\om,l}(\vp)$ along the potential deduced from $λ$ is the weight $F_{l}(λ)$. $F_{\om,l}(\vp)$ reduces to the Aubin energy when $l=0$ and the K-Energy map of Mabuchi when $l=1$. When $l\geq 2$ $F_{\om,l}(\vp)$ coincides (modulo lower order terms) with the functional $E_{\om,l-1}(\vp)$ introduced by X.X. Chen and G.Tian.

math.DG↗
arXiv2004

Analysis of Geometric Stability

We identify the difference between the CM polarisation and the Chow polarisation on the ``Hilbert scheme''. As a consequence, we give a numerical criterion for the CM stability as in Mumfords' G.I.T.. Also, we write down an explicit formula for the generalised futaki invariant interms of weights and multiplicities of the associated degeneration.

math.DG↗