Hassan Jolany

In this paper, by applying Greene-Shapere-Vafa-Yau semi-flat metric, we give a new proof of closed formula of Weil-Petersson metric on moduli space of Calabi-Yau varieties.

Existence of canonical metric on a canonical model of projective singular variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class(most of the varieties do not have definite first Chern class). There is a program which is known as Song-Tian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. In this paper, we apply Song-Tian program for mildly singular pair (X, D) via Log Minimal Model Program where D is a simple normal crossing divisor on X with conic singularities. We show that there is a unique C∞-fiberwise conical Kahler-Einstein metric on ( X, D) with vanishing Lelong number which is twisted by logarithmic Weil-Petersson metric and an additional term of Fujino-Mori [72] as soon as we have fiberwise KE-stability or Kawamata’s condition of Theorems 2.28, or 2.30(in C^0-case). In final we highlight that how the complete answer of this question can be reduced to CMA equation corresponding to fiberwise Calabi-Yau foliation(due to H.Tsuji) which is still open.

In this paper, we give a short proof of closed formula [9],[18] of logarithmic Weil-Peterssonmetric on moduli space of log Calabi-Yau varieties (if exists!) of conicand Poincare singularities and its connection with Bismut-Vergne localization formula.Moreover we give a relation between logarithmic Weil-Petersson metric andthe logarithmic version of semi Ricci flat metric on the family of log Calabi-Yaupairs with conical singularities. We highlight Bismut-Gillet-Soule fiberwise integralformula for logarithmic Weil-Petersson metric on moduli space of polarizedlog Calabi-Yau spaces by using fiberwise Ricci flat metric. In final we considerthe semi-positivity of singular logarithmic Weil-Petersson metric on the modulispace of log-Calabi-Yau varieties. Moreover, we show that Song-Tian-Tsuji measureis bounded along Iitaka fibration if and only if central fiber has log terminalsingularities and we consider the goodness of fiberwise Calabi-Yau metric in thesense of Mumford and goodness of singular Hermitian metric corresponding toSong-Tian-Tsuji measure. We also mention that the Song-Tian-Tsuji measure isbounded near origin if and only if after a finite base change the Calabi-Yau familyis birational to one with central fiber a Calabi-Yau variety with at worst canonicalsingularities.

In this paper, we define multi poly-Bernoulli polynomials using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. Furthermore, an explicit formula for certain Hurwitz-Lerch type multi poly-Bernoulli polynomials is established using the $r$-Whitney numbers of the second kind.

In this paper, we establish more identities of generalized multi poly-Euler polynomials with three parameters and obtain a kind of symmetrized generalization of the polynomials. Moreover, generalized multi poly-Bernoulli polynomials are defined using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. These are generalized further using the concept of Hurwitz-Lerch multiple zeta values.

In this paper we investigate special generalized Bernoulli polynomials with a,b,c parameters that generalize classical Bernoulli numbers and polynomials. The present paper deals with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a,b,c paramerers therefore seems to be a natural and important problem. By using the generalization of poly-Bernoulli polynomials with a,b,c parameters of negative index we define symmetrized generalization of poly-Bernoulli polynomials with a,b parameters of two variables and we prove duality property for them. Also by stirling numbers of the second kind we will find a closed formula for them. Furthermore we generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin integral, we define generalization of Arakawa-Kaneko Zeta functions with a,b parameters and we obtain an interpolation formula for the generalization of poly-Bernoulli numbers and polynomials with a,b parameters. Furthermore we present a link between this type of Zeta functions and Dirichlet series. By our interpolation formula, we will interpolate the generalization of Arakawa-Kaneko Zeta functions with a,b parameters.

In this paper, we establish more properties of generalized poly-Euler polynomials with three parameters and we investigate a kind of symmetrized generalization of poly- Euler polynomials. Moreover, we introduce a more general form of multi poly-Euler polynomials and obtain some identities parallel to those of the generalized poly-Euler polynomials.