# Can one determine the trace map for a nonsingular projective variety explicitly?

I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the canonical sheaf represents the relevant functor through some very non-explicit homological algebra gymnastics. The name of the map seems to suggest that I ought to be able to be able to write down some kind of matrix, hopefully in terms of homogeneous coordinates, and take its trace. Is this true (if not, why is it called a trace map)? How does one do this?

• Just to be clear, you're referring to the canonical isomorphism $H^n(X, \omega_X)\to k$, where $n$ is the dimension of $X$? If $k=\mathbb{C}$, this is just integration of top forms, as you probably know... – Daniel Litt Oct 19 at 18:24
• Yes, exactly that. – Sarah Griffith Oct 19 at 18:33
• The only situation where I know how to do this over general fields (not $\mathbb{C}$) is the case $X$ is a curve, where indeed this is given by some kind of (infinite-dimensional) trace, as explained by Tate here: numdam.org/article/ASENS_1968_4_1_1_149_0.pdf – Daniel Litt Oct 19 at 18:36