01121nas a2200121   4500008004300000245007400043210006900117260003700186520069600223100002200919700002200941856003600963       2010                        en_Ud  00aRiemann-Roch theorems and elliptic genus for virtually smooth schemes0 aRiemannRoch theorems and elliptic genus for virtually smooth sch  bMathematical Sciences Publishers3 aFor 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.1 aFantechi, Barbara1 aGöttsche, Lothar  uhttp://hdl.handle.net/1963/3888