We consider an n×n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=∞, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. When the system also has a Fuchsian singularity at z=0, we show under minimal vanishing conditions on the residue matrix at z=0 that isomonodromic deformations can be extended to the whole polydisc, including Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

1 aCotti, Giordano1 aDubrovin, Boris1 aGuzzetti, Davide uhttps://doi.org/10.1215/00127094-2018-005901258nas a2200133 4500008004100000245006300041210006300104260001000167520083800177100002001015700002001035700002101055856004801076 2018 en d00aLocal moduli of semisimple Frobenius coalescent structures0 aLocal moduli of semisimple Frobenius coalescent structures bSISSA3 aThere is a conjectural relation, formulated by the second author, between the enumerative geometry of a wide class of smooth projective varieties and their derived category of coherent sheaves. In particular, there is an increasing interest for an explicit description of certain local invariants, called monodromy data, of semisimple quantum cohomologies in terms of characteristic classes of exceptional collections in the derived categories. Being intentioned to address this problem, which, to our opinion, is still not well understood, we have realized that some issues in the theory of Frobenius manifolds need to be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence phenomenon.

1 aCotti, Giordano1 aDubrovin, Boris1 aGuzzetti, Davide uhttp://preprints.sissa.it/handle/1963/3530401295nas a2200133 4500008004100000245006800041210006800109300001200177490000700189520087800196100002001074700002101094856004601115 2017 eng d00aAnalytic geometry of semisimple coalescent Frobenius structures0 aAnalytic geometry of semisimple coalescent Frobenius structures a17400040 v063 aWe present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

1 aCotti, Giordano1 aGuzzetti, Davide uhttps://doi.org/10.1142/S201032631740004400449nas a2200085 4500008004100000245010400041210006900145100002000214856012900234 2016 eng d00aCoalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers0 aCoalescence Phenomenon of Quantum Cohomology of Grassmannians an1 aCotti, Giordano uhttps://www.math.sissa.it/publication/coalescence-phenomenon-quantum-cohomology-grassmannians-and-distribution-prime-numbers