This document is a corrigendum to our previous article “A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume”.
Let R be a complete discrete valuation ring of equal characteristic p > 0 . In this paper, we first generalize the notion of Hurwitz trees in mixed characteristic to Artin–Schreier covers ( ℤ / p -covers in characteristic p ) of a formal disc over R . It is a combinatorial-differential object with the shape of the dual graph of a cover’s semi-stable model, and is endowed with essential degeneration data, measured by Kato’s refined Swan conductors, of that cover. We then show how the existence of a deformation between two given Artin–Schreier covers is completely determined by the existence of certain Hurwitz trees. Finally, using the Hurwitz trees criteria, we improve a known result about the connectedness of the moduli space of Artin–Schreier curves of fixed genus.
Let α ∈ ℂ be an exponential period. This paper shows that the real and imaginary parts of α are up to signs volumes of sets definable without parameters in the o-minimal structure generated by the real exponential function and sin | [ 0 , 1 ] . This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary parts are up to signs volumes of ℚ -semi-algebraic sets. Furthermore, we define a notion of naive exponential period and compare it to the existing notions using cohomological methods. In particular, naive exponential periods are the same as periods of exponential Nori motives, which justifies that the definition of naive exponential periods singles out the correct set of complex numbers to be called exponential periods.
In this short note, we investigate noninvertible stochastic dynamical systems on the unit interval [ 0 , 1 ] . We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.
For any unbranched double covering of a compact Riemann surface, we study the associated character varieties that are unitary in the global sense, which we call GL n ⋊ 〈 σ 〉 -character varieties. We introduce k > 0 punctures on the surface, restrict the monodromies around the punctures to generic semi-simple conjugacy classes in GL n , and compute the E-polynomials of these character varieties using the character table of GL n ( q ) . The result is expressed as the inner product of certain symmetric functions. We are then led to a conjectural formula for the mixed Hodge polynomial, which is built out of (modified) Macdonald polynomials, their self-pairings, and self-pairings of wreath Macdonald polynomials.
