An obstruction to the existence of constant scalar curvature Kähler metrics (Q2496586)

The problem of existence of constant scalar curvature Kähler (cscK) metrics in a given Kähler class is a classical one in differential geometry. The simplest obstruction is provided by the non-vanishing of the Futaki invariant (a character on the Lie algebra of holomorphic vector fields). Further, \textit{G. Tian} [Invent. Math. 130, No.~1, 1--37 (1997; Zbl 0892.53027)] defined the notion of \(K\)-stability and showed this is an obstruction to the existence of cscK metrics. Recall that the definition of \(K\)-stability of a polarized Kähler manifold involves the so-called test-configurations with general fibre \((X,L)\). These are flat projective families of \(\mathbb Q\)-polarized schemes \(({\mathcal X},{\mathcal L})\to\mathbb C\) together with an action of \(\mathbb C^*\) on \(({\mathcal X}, {\mathcal L})\) covering the usual one on \(\mathbb C\) such that the fibre \(({\mathcal X}_t, {\mathcal L}_t)\) is isomorphic with \((X,L)\) for each \(t\neq 0\). The \(K\)-(semi)stability is assured by the (non-vanishing) positivity of a Donaldson-Futaki invariant \(F_1\) [see \textit{S. K. Donaldson}, J. Differ. Geom. 59, No. 3, 479--522 (2001; Zbl 1052.32017)] which can also be interpreted in terms of Mumford weight function in GIT. What is known for the moment is that \(K\)-semistability is a necessary condition for the existence of cscK metrics (Donaldson). Restricting to analytic test configuration leads to the notions of analytic \(K\)-(semi)stability. Finally, analytic \(K\)-polystability means analytic \(K\)-semistability together with the condition that any analytic test configuration with \(F_1=0\) is a product configuration. The main conjecture in the field is that for a polarized manifold \((X,L)\), \(K\)-polystability is equivalent with the existence of a cscK metric in \(c_1(L)\) (Yau-Tian-Donaldson). The new approach this paper brings in is considering a new notion, the slope stability, defined by analogy with vector bundles stability. To this end, the authors consider test configurations with non-normal central fibre. It is proven that slope semistability obstructs \(K\)-semistability and hence it obstructs the existence of cscK metrics. Very interesting examples of slope (semi)stable manifolds are then given in the paper. For instance, smooth polarized curves of genus \(g\geq 1\) are slope stable. Moreover, if \(C\) is a smooth curve of genus \(g\geq 1\) and \(E\to C\) is a vector bundle of rank \(\geq 2\) and \(L_m:={\mathcal O}_{\mathbb P(E)}(1)\otimes {\mathcal O}_C(m)\) is ample for large \(m\), then slope (semi/poly)stability of \((\mathbb P(E), L_m)\) implies Mumford slope (semi/poly)stability of \(E\). A sufficient condition for the slope stability of \((X,L)\) with respect to smooth subschemes is the numeric triviality of \(K_X\) or the ampleness of \(K_X\) and \(L\) being a multiple of \(K_X\). On the other hand, using blowups, numerous examples of slope unstable manifolds with respect to some polarizations are constructed. Namely, if \((X,L)\) is destabilized by \(Z\) and \(Y\) is the blowup of \(X\) along a centre disjoint from \(Z\), then for a polarisazion which make the exceptional set small, \(Y\) is destabilized by the proper transform of \(Z\). All these examples and counterexamples show that slope stability might be very effective in deciding whether a Kähler manifold admits cscK metrics or not.