On homothetic balanced metrics
Let \(g\) be a Kähler metric on a compact complex manifold \(M\). In the terminology of quantum mechanics a compact Kähler manifold \((M, g)\) is said to be quantizable if the Kähler form \(\omega\) associated to g is integral, i.e., there exists a holomorphic line bundle \(L\) over \(M\), called the \textit{quantum line bundle}, whose first Chern class equals the second De-Rham cohomology class of \(\omega\), that is \(c_1(L)=[\omega]_{dR}\). By Kodaira's theory this is equivalent to saying that \(M\) is a projective algebraic manifold and \(L\) is a positive (or ample) line bundle over \(M\). In terms of algebraic-geometric \(L\) is called a \textit{polarization} of \(M\), \(g\) a \textit{polarized} metric and the pair \((M, L)\) a \textit{polarized manifold}. Fix a polarization \(L\) over \(M\). Then there exists a Hermitian metric \(h\) on \(L\), uniquely defined up to the multiplication with a positive constant, such that its Ricci curvature equals \(\omega\). The pair \((L, h)\) is called a \textit{geometric quantization} of the Kähler manifold \((M, \omega)\). Let \(s_0, \dots , s_{N}\) be an orthonormal basis of \(H^0(L)\) (the space of global holomorphic sections of \(L\)) with respect to the \(L^2\) scalar product induced by \(\omega\). Consider the non-negative smooth function \(T_{g}\) on \(M\) given by \(T_{g} (x) =\sum_{j=0}^{N}h(s_j(x), s_j(x)).\) The metrics for which \(T_{g}\) is constant are called balanced in the terminology introduced by \textit{S. K. Donaldson} in [J. Differ. Geom. 59, No. 3, 479--522 (2001; Zbl 1052.32017)]. In the paper under consideration the authors study the set \({B}(L)\) consisting of all balanced metrics \(g_B\) and its subset \({B}_{g_B} = \{mg_{B}\text{ is balanced}| m \in \mathbb N^+\} \), for any \(g_B \in {B}(L)\). The quotient of \({B}(L)\) by the equivalence relation (which identifies two balanced metrics if they belong to the same cohomology class) is denoted by \({B}_c(L)\). The main results are: 1) If L is a polarization of a compact Kähler-Einstein manifold (M,g) with non-positive scalar curvature, then \({B}_c(L)\) consists of infinitely many balanced metrics such that \({B}_{g_B}\) is finite for any \(g_B \in {B}(L)\). 2) If \(g_{B}\) is a Kähler-Einstein metric on a toric manifold \(M\) of dimension \(\leq 4\) and \(L = K^{*}\) is the anticanonical bundle over \(M\), then the cardinality of \({B}_c(L)\) is finite. There exists \(g_B \in {B}(L)\) such that \({B}_{g_B}\) is infinite iff M is either a projective space or a product of projective spaces. 3) In the case of constant scalar curvature metrics found in [\textit{C. Arezzo} and \textit{F. Pacard}, Acta Math. 196, No. 2, 179--228 (2006; Zbl 1123.53036); Ann. Math. (2) 170, No. 2, 685--738 (2009; Zbl 1202.53069)], \({B}_{g_B}\) is finite. The proofs of these results are based on Tian-Yau-Zelditch expansion and the the theory of projectively induced Kähler metrics developed by \textit{E. Calabi} [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)].
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