Cartan Determinants in algebraic geometry

Abstract

We formulate a dg-categorical analogue of the Cartan determinant conjecture using numerical Grothendieck groups. Let $D$ be a smooth proper dg category over an algebraically closed field $k$. The Euler pairing on $K_0(D)$ descends to a nondegenerate bilinear form on the numerical Grothendieck group $K_{\mathrm{num}}(D)$, and hence defines a numerical Cartan determinant $\det M_{\chi }(D)$. While the classical Cartan determinant conjecture predicts that $\det M_{\chi }(A)=1$ for finite-dimensional smooth algebras $A$, this equality is no longer expected for general smooth proper dg categories. However, we propose the following positivity question: $\det M_{\chi }(D)>0.$ Our main result proves this positivity when $D=\mathrm{Perf}(X)$ for a large class of smooth projective varieties $X$. More precisely, we show that $\det M_{\chi }(X)>0$ if either $\dim X$ is odd, or $X$ has even dimension and satisfies the numerical Hodge standard conjecture.