A pair of a smooth variety and a regular function is called a Landau-Ginzburg (LG) model. For a LG model there is a notion of matrix factorizations. They are -curved 2-periodic complexes on . They appeared in the study of the singularity of the hypersurface of defined by and homological mirror symmetry for smooth Fano varieties.

In this talk we show a Hirzebruch-Riemann-Roch type formula for matrix factorizations: it is an explicit formula for the Euler characteristic of the Hom space between matrix factorizations, in terms of their Chern characters. When time permits, we also report a joint work with Dongwook Choa and Bhamidi Sreedhar: Hochschild-Kostant-Rosenberg type isomorphism and Hirzebruch-Riemann-Roch type formula for a LG orbifold model.

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