A Proof Via the Seiberg-Witten Moduli Space of Donaldson’s Theorem on Smooth $4$-Manifolds with Definite Intersection Forms

Mikhail Katz

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Moduli of smoothness of functions and their derivatives

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Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for $L_{p} (T)$ and $L_{p} [- 1, 1]$ for 0 < p < ∞ using the moduli of smoothness $ω^{r} {(f, t)}_{p}$ and $ω_{φ}^{r} {(f, t)}_{p}$ respectively.

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In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If $r$ is the Cartier index of the fibre, it was expected that $12 r$ would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in $r$ , we provide a bound $N (r)$ and an example where the smallest integer that clears the denominators of the moduli part is $N (r) / r$ . Moreover we prove that even locally the denominators...

The KSBA compactification for the moduli space of degree two $K 3$ pairs

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Journal of the European Mathematical Society

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Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X, H)$ consisting of a degree two $K 3$ surface $X$ and an ample divisor $H$ . Specifically, we construct and describe explicitly a geometric compactification ${\bar{P}}_{2}$ for the moduli of degree two $K 3$ pairs. This compactification...

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Journal of the European Mathematical Society

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In this paper we consider questions of the following type. Let $k$ be a base field and $K / k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$ ), what is the least transcendence degree of a field of definition of $X$ over the base field $k$ ? In other words, how many independent parameters are needed to define $X$ ? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete...

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Soluble Groups with Many Černikov Quotients

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Si studiano i gruppi risolubili non di Černikov a quozienti propri di Černikov. Nel caso periodico tali gruppi sono tutti e soli i prodotti semidiretti $H\ltimes N$ con $N$ $p$ -gruppo abeliano elementare infinito e $H$ gruppo irriducibile di automorfismi di $N$ che sia infinito e di Černikov. Nel caso non periodico invece si riconduce tale studio a quello dei moduli a quozienti propri artiniani su un gruppo risolubile finito, e si fornisce una caratterizzazione di tali moduli.

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Multiple end solutions to the Allen-Cahn equation in $ℝ^{2}$

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An entire solution of the Allen-Cahn equation $Δ u = f (u)$ , where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$ , e.g. $f (u) = u (u^{2} - 1)$ , is called a $2 k$ end solution if its nodal set is asymptotic to $2 k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $- 1$ ) like the one dimensional, odd, heteroclinic solution $H$ , of $H^{''} = f (H)$ . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space...

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Displaying similar documents to “A Proof Via the Seiberg-Witten Moduli Space of Donaldson’s Theorem on Smooth 4 -Manifolds with Definite Intersection Forms”

Displaying similar documents to “A Proof Via the Seiberg-Witten Moduli Space of Donaldson’s Theorem on Smooth $4$ -Manifolds with Definite Intersection Forms”