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Register a new userGeometirc Arveson-Douglas Conjecture - Decomposition of Varieties
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In this paper, we prove the Geometric Arveson-Douglas Conjecture for a special case which allow some singularity on $partial{mathbb{B}_n}$. More precisely, we show that if a variety can be decomposed into two varieties, each having nice properties and intersecting nicely with $partialmathbb{B}_n$, then the Geometric Arveson-Douglas Conjecture holds on this variety. We obtain this result by applying a result by Suarez, which allows us to localize the problem. Our result then follows from the simple case when the two varieties are intersection of linear subspaces with $mathbb{B}_n$.
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In this paper we introduce techniques from complex harmonic analysis to prove a weaker version of the Geometric Arveson-Douglas Conjecture for complex analytic subsets that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra $mathcal{T}(L^{infty})$, which implies the essential normality of the quotient module. Combining some other techniques we actually obtain the $p$-essential normality for $p>2d$, where $d$ is the complex dimension of the analytic subset. Finally, we show that our results apply for the closure of a radical polynomial ideal $I$ whose zero variety satisfies the above conditions. A key technique is defining a right inverse operator of the restriction map from the unit ball to the analytic subset generalizing the result of Beatrouss paper $L^p$-estimates for extensions of holomorphic functions.
Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of $n$-fold tensor products of Drury-Arveson spaces.
We show that for every pair of matrices (S,P), having the closed symmetrized bidisc $Gamma$ as a spectral set, there is a one dimensional complex algebraic variety $Lambda$ in $Gamma$ such that for every matrix valued polynomial f, the norm of f(S,P) is less then the sup norm of f on $Lambda$. The variety $Lambda$ is shown to have a particular determinantal representation, related to the so-called fundamental operator of the pair (S,P). When (S,P) is a strict $Gamma$-contraction, then $Lambda$ is a distinguished variety in the symmetrized bidisc, i.e., a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.
The Drury-Arveson space $H^2_d$, also known as symmetric Fock space or the $d$-shift space, is a Hilbert function space that has a natural $d$-tuple of operators acting on it, which gives it the structure of a Hilbert module. This survey aims to introduce the Drury-Arveson space, to give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and to describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.
The Shafarevich-Tate group $W (mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (mathscr{A})cong Cl~(Lambda)oplus Cl~(Lambda)$ or $W (mathscr{A})cong left(mathbf{Z}/2^kmathbf{Z}right) oplus Cl_{~mathbf{odd}}~(Lambda)oplus Cl_{~mathbf{odd}}~(Lambda)$, where $Cl~(Lambda)$ is the ideal class group of a ring $Lambda$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Lambda)$. The case of elliptic curves with complex multiplication is considered in detail.
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