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Bounds for sectional genera of varieties invariant under Pfaff fields. (English) Zbl 1281.32027
The abstract says: “We establish an upper bound for the sectional genus of varieties which are invariant under Pfaff fields on projective spaces”. A more detailed summary should at least (assuming the reader is familiar with the nomenclature) include the statement of their results:
Theorem. Let \(X\) be a nonsingular projective variety of dimension \(m\) which is invariant under a Pfaff field \({\mathcal F}\) of rank \(k\) on \({\mathbb P}^n\); assume that \(m \geq k\). If the tangent bundle \(\Theta_{X}\) is stable, then \[ \frac{2g(X, {\mathcal O}_X(1))-2}{\deg(X)} \leq \frac{\deg({\mathcal F})-k}{\binom{m-1}{k-1}}+m-1. \]
Theorem. Let \(X\subset {\mathbb P}^n\) be a Gorenstein projective variety nonsingular in codimension \(1\), which is invariant under a Pfaff field \({\mathcal F}\) on \({\mathbb P}^n\) whose rank is equal to the dimension of \(X\). Then \[ \frac{2g(X, {\mathcal O}_X(1))-2}{\deg(X)} \leq \deg({\mathcal F}) - 1. \]
This result generalizes the bounds in [A. Campillo et al., J. Lond. Math. Soc., II. Ser. 62, No. 1, 56–70 (2000; Zbl 1040.32027)].

MSC:
32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
58A17 Pfaffian systems
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Full Text: Euclid
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