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Abstract The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$where$$f: X \to {\Bbb R}$$,Xa set, finite or infinite, andKand$$\mu $$denote a suitable kernel and a measure, respectively. Given a connected ordered graphGonnvertices, consider the multi-linear form$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$where$${\mathcal E}(G)$$is the edge set ofG. Define$$\Lambda _G(p_1, \ldots , p_n)$$as the smallest constant$$C>0$$such that the inequality(0.1)$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$holds for all nonnegative real-valued functions$$f_i$$,$$1\le i\le n$$, onX. The basic question is, how does the structure ofGand the mapping properties of the operator$$T_K$$influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case$$X={\Bbb F}_q^d$$, thed-dimensional vector space over the field withqelements,$$K(x^i,x^j)$$is the indicator function of the sphere evaluated at$$x^i-x^j$$, and connected graphsGwith at most four vertices.
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A virtual PGL–SL correspondence for projective surfaces
Abstract For a smooth projective surface$$X$$satisfying$$H_1(X,\mathbb{Z}) = 0$$and$$w \in H^2(X,\mu _r)$$, we study deformation invariants of the pair$$(X,w)$$. Choosing a Brauer–Severi variety$$Y$$(or, equivalently, Azumaya algebra$$\mathcal{A}$$) over$$X$$with Stiefel–Whitney class$$w$$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on$$Y$$constructed by Yoshioka (or, equivalently, moduli spaces of$$\mathcal{A}$$-modules of Hoffmann–Stuhler). We show that the invariants do not depend on the choice of$$Y$$. Using a result of de Jong, we observe that they are deformation invariants of the pair$$(X,w)$$. For surfaces with$$h^{2,0}(X) \gt 0$$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker–Maruyama–Simpson moduli spaces of stable sheaves on$$X$$. This can be seen as a$${\rm PGL}_r$$–$${\rm SL}_r$$correspondence. As an application, we express$${\rm SU}(r) / \mu _r$$Vafa–Witten invariants of$$X$$in terms of$${\rm SU}(r)$$Vafa–Witten invariants of$$X$$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on$$X$$with given division algebra at the generic point.
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- Award ID(s):
- 2401484
- PAR ID:
- 10586664
- Publisher / Repository:
- Cambridge
- Date Published:
- Journal Name:
- Moduli
- Volume:
- 2
- Issue:
- e5
- ISSN:
- 2949-7647
- Page Range / eLocation ID:
- 1-41
- Subject(s) / Keyword(s):
- Vafa-Witten invariants, S-duality
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/mod.2024.13
