Stability Conditions and Positivity of Invariants of Fibrations

Abstract

We study three methods that prove the positivity of a natural numerical invariant associated to 1-parameter families of polarized varieties. All these methods involve different stability conditions. In dimension 2 we prove that there is a natural connection between them, related to a yet another stability condition, the linear stability. Finally we make some speculations and prove new results in higher dimension.

Generalized Clifford–Severi inequality and the volume of irregular varieties

“…We remind briefly the method in this simplified version and refer to [2], [13], [23] and [29] for details. The construction holds in general for Q-Cartier Weil divisors but for simplicity we state it for Cartier divisors, which is the case we will use.…”

“…In fact, f -positivity of ω f for dimensions less or equal to n implies a weaker inequality for the slope K n f ≥ 2n! χ f from which one can deduce the Severi inequality for L = ω X (see [2] Proposition 5.8, for a proof of this statement). We prove a converse, namely, that the Severi inequality for L = ω f implies the above slope inequality when b = g(B) = 0 and a slightly weaker result when b ≥ 1.…”

“…Given a fibred variety over a smooth curve, f : X −→ B with general fibre F , and a line bundle L on X, slope inequalities relate the invariants of (X, L), (F, L |F ) and B. The best slope inequalities hold when some stability properties hold for L (see [2] for a survey on this topic), and then a slope inequality, which we call f -positivity of L, follows…”

Stability and Singularities of Relative Hypersurfaces

“…In [2] we have seen that the known methods to prove f -positivity all need to assume some stability condition. However we do not need any kind of stability assumption for proving the above results for hypersurfaces.…”

“…However we do not need any kind of stability assumption for proving the above results for hypersurfaces. Some of the methods described in [2], in particular the one due to Cornalba-Harris and Bost (Theorem 3.1) can thus be used backwards in this context to prove an instability result (Theorem 3.2): Theorem 1.3. Let E be a µ-unstable sheaf.…”