Abstract: After reviewing some results from last week's talk concerning moment-angle complexes $\mathcal{Z}_K$ and their cohomology rings, I will describe some further structure on $H^*(\mathcal{Z}_K)$ given by cohomology operations induced by the standard torus action. Under the identification of $H^*(\mathcal{Z}_K)$ with the Koszul homology of the Stanley-Reisner ring of $K$, these operations assemble to give an explicit differential on the minimal free resolution of the Stanley-Reisner ring. Using this topological interpretation of the minimal free resolution, we give simple algebraic and combinatorial characterizations of equivariant formality for torus actions on moment-angle complexes. This is joint work with Benjamin Briggs.

Abstract: A compact set $X\subset\mathbb C^n$ is said to be rationally convex if for every point $z\not\in X$ there is a polynomial $P$, depending on $z$, so that $P(z)=0$ but $P^{-1}(0)\cap X=\varnothing$. In view of the Oka-Weil theorem, any function holomorphic on a rationally convex compact $X$ can be approximated uniformly on $X$ by rational functions with poles off $X$. A totally real manifold $M$ is one whose tangent space has no complex structure, i.e., $J(T_pM)\cap T_pM=\{0\}$ for all $p\in M$. By a classical result of Duval-Sibony, a totally real manifold $M$ in $\mathbb{C}^n$ is rationally convex if and only if there exists a Kähler form $dd^c\varphi$ for which $M$ is isotropic. Under a mild technical assumption, we generalize this necessary and sufficient condition to the setting of totally real sets (zero loci of strictly plurisubharmonic functions).