Yuuki Shiraishi

Starting from the Weierstrass elliptic function, we study the associated Frobenius structure, incorporating the perspective of derived categories, particularly that of homological mirror symmetry. Given a deformation of the Weierstrass elliptic function, we construct a primitive form normalized to be compatible with the period map for integral cycles, and obtain a Frobenius structure whose Frobenius potential is defined over the rational numbers. We also construct a Frobenius structure using elliptic Weyl group invariants (as opposed to Jacobi group invariants), and establish an isomorphism between these two Frobenius structures. We further examine the relationship between the degree of the Lyashko--Looijenga map modulo the modular group and the number of full exceptional collections up to the braid group action and translations, as well as the associated Gamma-integral structure.

This paper calculates the number of full exceptional collections modulo an action of a free abelian group of rank one for an abelian category of coherent sheaves on an orbifold projective line with a positive orbifold Euler characteristic, which is equivalent to the one of finite dimensional modules over an extended Dynkin quiver of ADE type by taking their derived categories. This is done by a recursive formula naturally generalizing the one for the Dynkin case by Deligne whose categorical interpretation is due to Obaid--Nauman--Shammakh--Fakieh--Ringel. Moreover, the number coincides with the degree of the Lyashko--Looijenga map of the Frobenius manifold for the orbifold projective line, which hints a consistency in some problems in Bridgeland's stability conditions and mirror symmetry.

We show the uniqueness and existence of the Euler form for a simply-laced generalized root system. This enables us to show that the Coxeter element for a simply-laced generalized root system is admissible in the sense of R.~W.~Carter. As an application, the isomorphism classes of simply-laced generalized root systems with positive definite Cartan forms are classified by Cartar's admissible diagrams associated to their Coxeter elements.