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Let ( X , h ) be a compact and irreducible Hermitian complex space of complex dimension m . In this paper we are interested in the Dolbeault operator acting on the space of L 2 sections of the canonical bundle of reg ( X ) , the regular part of X . More precisely let d ‾ m , 0 : L 2 Ω m , 0 ( reg ( X ) , h ) → L 2 Ω m , 1 ( reg ( X ) , h ) be an arbitrarily fixed closed extension of ∂ ‾ m , 0 : L 2 Ω m , 0 ( reg ( X ) , h ) → L 2 Ω m , 1 ( reg ( X ) , h ) where the domain of the latter operator is Ω c m , 0 ( reg ( X ) ) . We establish various properties such as closed range of d ‾ m , 0 , compactness of the inclusion D ( d ‾ m , 0 ) ↪ L 2 Ω m , 0 ( reg ( X ) , h ) where D ( d ‾ m , 0 ) , the domain of d ‾ m , 0 , is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge–Kodaira Laplacian d ‾ m , 0 ⁎ ∘ d ‾ m , 0 with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to d ‾ m , 0 ⁎ ∘ d ‾ m , 0 , with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge–Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal. As an application, we deduce that the secant variety to a general canonical curve of genus at least 7 is normal. [ABSTRACT FROM AUTHOR]

We study the holomorphic equivalence problem for finite type hypersurfaces in C 2 . We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in C 2 . [ABSTRACT FROM AUTHOR]

We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev–Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow. [ABSTRACT FROM AUTHOR]

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez-Leclerc from the Kronecker and Dynkin quiver case to the acyclic case. The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization. [ABSTRACT FROM AUTHOR]

Abstract: We classify open maximal subalgebras of all infinite-dimensional linearly compact simple Lie superalgebras. This is applied to the classification of infinite-dimensional Lie superalgebras of vector fields, acting transitively and primitively in a formal neighborhood of a point of a finite-dimensional supermanifold. [Copyright &y& Elsevier]