Generalized virtual polyhedral and cohomology of torus manifolds

Askold Khovanskii, University of Toronto

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Passcode: 114700

About 30 years ago, thinking about BKK theorem and smooth toric varieties, Aleksandr Pukhlikov and I introduced virtual polyhedra and developed the theory of finite additive polynomial measures on such objects. One of our results is a description of the cohomology ring of smooth toric varieties in terms of volumes of virtual polyhedra. Recently, thinking about torus manifolds, together with Leonid Monin and Ivan Limonchenko, we introduced generalized virtual polyhedra as well as smooth polynomial measures on them. Repeating Khovanskii–Pukhlikov construction we described the cohomology ring of torus manifolds in terms of volumes of such objects. We also discovered an analog of BKK theorem for torus manifolds. Similar result was obtained by Mikiya Masuda and Anton Ayzenberg. On my talk I will concentrate on geometry and on homotopical topology related to generalized virtual polyhedra.