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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
3
votes
Action of $\mathbb C^*$ on a closed orbit of a torus
In fact $X$ is a toric variety. "In general" the automorphism group of this variety is the torus himself. But in some cases it is not. For instance if $\varphi$ is the diagonal action of $\bf {C^*} ^n …
5
votes
Euler characteristic of a surface in $\mathbb{R}^3.$
You can assume that the $z$ function is Morse, compute the critical points (solve a Lagrange multiplier problem) and their indices (compute a second derivative), and remember that $\chi (S)= \sum _{c …