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The identities for elliptic gamma functions discovered by Felder and Varchenko  are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in -dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three. (It is a stack.) Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve
We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative -space for some . This is a noncommutative version of Rosenthal's result for commutative -spaces. Similarly for , an infinite-dimensional subspace of a noncommutative -space either contains or embeds in for some . The novelty in the noncommutative setting is a double-sided change of density
Let be a polynomial, where are real-analytic functions in an open subset of . If, for any , the polynomial has only real roots, then we can write those roots as locally Lipschitz functions of . Moreover, there exists a modification (a locally finite composition of blowups with smooth centers) such that the roots of the corresponding polynomial , can be written locally as analytic functions of . Let , be an analytic family of symmetric matrices, where is open in . Then there exists a modification such that the corresponding family can be locally diagonalized analytically (i.e., we can choose locally a basis of eigenvectors in an analytic way). This generalizes Rellich's well-known theorem (see ) from 1937 for -parameter families. Similarly, for an analytic family , of antisymmetric matrices, there exists a modification such that we can find locally a basis of proper subspaces in an analytic way
We show that the semistable conjecture of Fontaine and Jannsen (see ) is true for proper, vertical, fine, and saturated log-smooth families with reduction of Cartier type (e.g., proper schemes with simple semistable reduction). We derive it from Suslin's comparison theorem [31, Corollary 4.3] between motivic cohomology and étale cohomology. This gives a new proof of the semistable conjecture showing motivic character of p-adic period maps
Let be a field not of characteristic two, and let be a set consisting of almost all rational primes invertible in . Suppose that we have a variety and strictly compatible system of constructible -sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of is a subgroup of a corresponding isometry group over , and we say that it has big monodromy if it contains the derived subgroup . We prove a theorem that gives sufficient conditions for to have big monodromy. We apply the theorem to explicit systems arising from the middle cohomology of families of hyperelliptic curves and elliptic surfaces to show that the monodromy is uniformly big as we vary and the system. We also show how it leads to new results for the inverse Galois problem