Reading a proof of the Schwarz lemma for the Kähler-Ricci flow from p22 of these lecture notes. I am confused as to what they mean by taking $$\inf _{x \in M} \{\hat{R}_{i \bar i j \bar j}(x) \mid \{\partial_{z^{1}}, \ldots, \partial_{z^{n}} \} \text{ is orthornormal w.r.t. } \hat{g} \text{ at } x, \ i, j = 1, \ldots, n\} \ .$$ Is this lower bound even invariantly defined on the manifold?
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Their idea is correct, but its formulation (and the chosen notation) is indeed a bit sloppy. What they seem to do, in reality, is to define
$$ - \hat C = \inf _{x \in M} \inf \{ \hat R (e_i, \bar {e_i}, e_j, \bar {e_j}) \mid \{e_1, \bar {e_1}, \ldots, e_n, \bar {e_n} \} \text{ is orthornormal w.r.t. } \hat{g} \text{ at } x, \ i, j = 1, \ldots, n\} \ .$$
The inner infimum exists, because $i,j$ take a finite number of values, and the orthonormal frames $\{ e_1, \bar {e_1}, \dots, e_n, \bar {e_n} \}$ at $x$ form a space homeomorphic to $O(2n)$ which is compact.
They tacitly accept that this inner infimum will be a continuous function of $x$ (up to the reader to prove this!), therefore the outer infimum will be finite ($M$ is assumed compact).
Finally, in equation (2.22) they just work at some $x$, taking $\{ e_1, \bar {e_1}, \dots, e_n, \bar {e_n} \}$ to be precisely the frame $\{ \partial_1, \bar \partial_1, \dots, \partial_n, \bar \partial_n \}$ corresponding to the normal coordinates at $x$ with respect to $\hat g$.
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$\begingroup$ I see, thanks. Just one more question. I am also looking at a similar computation in page 17 of math.iisc.ac.in/~vvdatar/Lecture_Notes/Calabi_Yau.pdf where an upper bound $B$ on the holomorphic sectional curvature is defined to be $\hat{R}_{i \bar j k \bar l} \leq B(\hat{g}_{i \bar j} \hat{g}_{kl} + \hat{g}_{i \bar l} \hat{g}_{k \bar j})$. How are these two kinds of bounds related? $\endgroup$– shiyuCommented Dec 28, 2021 at 20:40
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$\begingroup$ I don't really get your question: why should these two bounds be related? In the second reference, the authors explicitly show that $B$ can be taken in at least two ways (for instance, the supremum over $p \in M$ of the supremum of $(\nabla_u \nabla_v \hat g) (e,f)$ over all systems $\{u,v,e,f\}$ of orthonormal vectors at $p$). I don't see any relationship between these bounds, but I am not an expert in complex geometry. $\endgroup$– Alex M.Commented Dec 28, 2021 at 21:09
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$\begingroup$ @shiyu: In the notes that you have linked, there is a typo. The phrase 'holomorphic sectional curvature' should be replaced by 'bi-sectional curvature'. See the answer of Yang-Mills for more details - mathoverflow.net/questions/106743/… . With this correction you just have to replace the inf with sup to get the upper bound in the notes that you have linked. $\endgroup$ Commented Jan 12, 2022 at 11:33