Birational Transformations on Irreducible Compact Hermitian Symmetric Spaces.

The article focuses on explicitly constructing a sequence of blow-ups and blow-downs that transform an irreducible compact Hermitian symmetric space \(X\) into a projective space of the same dimension, thereby resolving a birational map introduced by Landsberg and Manivel. The centers of these blow-ups on \(X\) are defined via loci of chains of minimal rational curves related to the Białynicki-Birula decomposition under a \(\mathbb{C}^*\)-action, while those on the projective space arise from the variety of minimal rational tangents (VMRT) of \(X\) and its higher secant varieties. The main theorem establishes that for \(X\) of rank \(r \geq 2\), there exist \(r-1\) successive blow-ups and blow-downs along smooth centers that yield an isomorphism between the resulting blow-up spaces of \(X\) and \(\mathbb{P}^n\). The article further describes the stratifications of these centers, their singularities, and their smoothing via blow-ups, and relates the construction to the Białynicki-Birula decomposition and the polysphere theorem, providing explicit examples and uniform proofs independent of classification.