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Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $\tilde V$. We compare modules over the ring $O (\tilde V)$ of regular functions on $\tilde V$ with modules over the ring $C^\infty (V)$ of... more
Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $\tilde V$. We compare modules over the ring $O (\tilde V)$ of regular functions on $\tilde V$ with modules over the ring $C^\infty (V)$ of smooth complex valued functions on V. Under a mild condition on the tangent spaces, we prove that $C^\infty (V)$ is flat as a module over $O (\tilde V)$. From this we deduce a comparison theorem for the Hochschild homology of finite type algebras over $O (\tilde V)$ and the Hochschild homology of similar algebras over $C^\infty (V)$. We also establish versions of these results for functions on $\tilde V$ (resp. V) that are invariant under the action of a finite group G. When V is compact, we check that $C^\infty (V)$ has finite rank as module over $C^\infty (V)^G$.
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Let k be a perfect field and let X⊂ℙ^N be a hypersurface of degree d defined over k and containing a linear subspace L defined over an algebraic closure k with codim_ℙ^NL=r. We show that X contains a linear subspace L_0 defined over k... more
Let k be a perfect field and let X⊂ℙ^N be a hypersurface of degree d defined over k and containing a linear subspace L defined over an algebraic closure k with codim_ℙ^NL=r. We show that X contains a linear subspace L_0 defined over k with codim_ℙ^NL≤ dr. Furthermore, we propose an explicit algorithm for finding L_0 (with a worse estimate on codimension) from L and its Galois conjugates. We also prove a similar result for a certain version of the Schmidt rank for quartic polynomials.
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This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give... more
This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting examples of such algebras. In particular, we construct a quantum vertex operator algebra from a rational, trigonometric, or elliptic R-matrix, which is a quantum deformation of the affine vertex operator algebra. The simplest vertex operator in this algebra is the quantum current of Reshetikhin and Semenov-Tian-Shansky.
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Given a hypersurface $X\subset \mathbb{P}^{N+1}_{\mathbb{C}}$ Dimca gave a proof showing that the cohomologies of X are the same as the projective space in a range determined by the dimension of the singular locus of X. We prove the... more
Given a hypersurface $X\subset \mathbb{P}^{N+1}_{\mathbb{C}}$ Dimca gave a proof showing that the cohomologies of X are the same as the projective space in a range determined by the dimension of the singular locus of X. We prove the analog of Dimca's result case when $\mathbb{C}$ is replaced with an algebraically closed field of finite characteristic and singular cohomology is replaced with $\ell$-adic etale cohomology. The Weil conjectures allow relating results about eatle cohomology to counting problems over a finite field. Thus by applying this result, we are able to get a relationship between the algebraic properties of certain polynomials and the size of their zero set.
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This paper is a continuation of [EK1-4]. The goal of this paper is to define and study the notion of a quantum vertex operator algebra (VOA) in the setting of the formal deformation theory and give interesting examples of such algebras.... more
This paper is a continuation of [EK1-4]. The goal of this paper is to define and study the notion of a quantum vertex operator algebra (VOA) in the setting of the formal deformation theory and give interesting examples of such algebras. Our definition of a quantum VOA is based on the ideas of the paper [FrR]. The first chapter of our paper is devoted to the general theory of quantum VOAs. For simplicity we consider only bosonic algebras, but all the definitions and results admit a straightforward generalization to the super-case. We start with the version of the definition of a VOA in which the main axiom is the locality (commutativity) axiom. To obtain a quantum deformation of this definition, we replace the locality axiom with the S-locality axiom, where S is a shift-invariant unitary solution of the quantum Yang-Baxter equation (the other axioms are unchanged). We call the obtained structure a braided VOA. However, a braided VOA does not necessarily satisfy the associativity prop...
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This article sets the foundations of a theory of perverse sheaves on arc spaces as it was conjectured by Feigin-Frenkel in 1990. We show a structure theorem which says that arc spaces are locally products of something finite dimensional... more
This article sets the foundations of a theory of perverse sheaves on arc spaces as it was conjectured by Feigin-Frenkel in 1990. We show a structure theorem which says that arc spaces are locally products of something finite dimensional times an infinite dimensional affine space for the pro-smooth topology. In particular, it answers a conjecture of Kollar and Nemethi. We then introduce a new topos, which allows us to define a bounded constructible category of l-adic sheaves equipped with a t-structure and with a Verdier dual. Finally we define an intersection complex whose fibers are given by the finite dimensional models obtained using Drinfeld-Grinbeg-Kazhdan's theorem.
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This article sets the foundations of a theory of perverse sheaves on arc spaces as it was conjectured by Feigin-Frenkel in 1990. We show a structure theorem which says that arc spaces are locally products of something finite dimensional... more
This article sets the foundations of a theory of perverse sheaves on arc spaces as it was conjectured by Feigin-Frenkel in 1990. We show a structure theorem which says that arc spaces are locally products of something finite dimensional times an infinite dimensional affine space for the pro-smooth topology. In particular, it answers a conjecture of Kollar and Nemethi. We then introduce a new topos, which allows us to define a bounded constructible category of l-adic sheaves equipped with a t-structure and with a Verdier dual. Finally we define an intersection complex whose fibers are given by the finite dimensional models obtained using Drinfeld-Grinbeg-Kazhdan's theorem.
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In this paper we construct explicitly the quantization of Lie bialgebras of a finite dimensional simple Lie algebra. by reducing the problem of quantization of the algebra of $\g$-valued functions on a curve with many punctures to the... more
In this paper we construct explicitly the quantization of Lie bialgebras of a finite dimensional simple Lie algebra. by reducing the problem of quantization of the algebra of $\g$-valued functions on a curve with many punctures to the case of one puncture