Kenichi Yoshikawa

Last Update 2018/08/08
My current research subject is holomorphic torsion invariant of algebraic manifolds with trivial canonical line bundle. One of the typical examples of such invariants is the one for K3 surfaces with involution, which I introduced in 2004. There are 75 deformation types of K3 surfaces with involution, and correspondingly we have 75 moduli spaces. On each moduli space, the invariant is given by the Petersson norm of a Borcherds product and its explicit formula is known. It is an interesting open problem of understanding the geometric meaning of the corresponding elliptic modular form. Another typical example is the so called BCOV invariant for CalabiYau threefolds, which was introduced in 1994 by BershadskyCecottiOoguriVafa as the counterpart in Bmodel of genusone instanton numbers of CalabiYau threefolds. This invariant is very important because the mirror symmetry at genusone is formulated as the equality of two functions on the moduli spaces, where one is obtained as the generating function of genusone instanton numbers and the other is obtained as the BCOV invariant. I computed the BCOV invariant in some classical examples (mirror quintics, BorceaVoisin threefolds). Recently I extend the construction of BCOV invariants to abelian CalabiYau orbifolds of dimension three. Besides mirror symmetry at genusone, I am interested in the problem such as the birational invariance of BCOV invariants and the comparison of BCOV invariants between CalabiYau orbifolds and their crepant resolutions. Very recently, I am studying holomorphic torsion invariant of higher dimensional manifolds.
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