We prove that every homomorphism O ζ → O ζ , with E and F Banach spaces and ζ ∈ C, is induced by a Hom(E,F )-valued holomorphic germ, provided that 1 ≤ m < ∞. A similar structure theorem is obtained for the homomorphisms of type O ζ → Sζ , where Sζ is a stalk of a coherent sheaf of positive mζ -depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheave...

This talk presents some new notions of the theory of complex algebraic curves which have appeared as algebraic tools in string theory (see [4] for more details). In a sense, we have materialized non-existed complex powers of invertible sheaves on algebraic curves introduced at the level of the Atiyah algebras of invertible sheaves by Beilinson and Schechtman [1]. The Atiyah algebra AL in the ca...

Let κ = Fq be a finite field of characteristic p > 0, and k be a fixed algebraic closure of κ. We fix a prime ` 6= p, and an isomorphism τ : Q` → C. Whenever we want to denote something (e.g. scheme, sheaf, morphism, etc.) defined over κ, we will put a subscript 0 (e.g. X0 is a scheme over κ, F0 is a Weil sheaf defined over X0, etc.), and when the subscript is dropped, this means the correspond...

Let Y, Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf OX to a sheaf of noncommutative algebras and of the sheaves OY and OZ to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf TorX q (OY ,...

Let Y, Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf OX to a sheaf of noncommutative algebras and of the sheaves OY and OZ to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf TorX q (OY ,...

Let $\{\Lambda ^\infty _t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$ that arise as slices singular Legendrian $\Lambda _I^\infty \subset S^*M \times T^*I$ . $\mathcal {C}_t = Sh(M, \Lambda _t)$ the differential graded derived category constructible sheaves on $M$ with support at infinity contained _t$ We prove if embeds into Liouville hypersurfaces, the...

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the ...