TY - JOUR T1 - Riemann-Roch theorems and elliptic genus for virtually smooth schemes JF - Geom. Topol. 14 (2010) 83-115 Y1 - 2010 A1 - Barbara Fantechi A1 - Lothar Göttsche AB - For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves. PB - Mathematical Sciences Publishers UR - http://hdl.handle.net/1963/3888 U1 - 821 U2 - Mathematics U3 - Mathematical Physics ER -