Dependencies

Suppose that \(R\) is factor-filtered. Then

is a subring of \(R\) which is in fact a field.

The ring \(H^0(X, \mathcal{O}_X)\) is a field.

Let \(F\) be a nonzero vector bundle on \(X\) whose HN slopes are all nonnegative. Then \(F\) is generated by global sections and \(H^1(X, F) = 0\).

For any \(\lambda \in \mathbb {Q}\), there is at most one isomorphism class of stable vector bundles on \(X\) of slope \(\lambda \).

By an abstract complete curve, we mean an abstract complete curve together with a degree function from closed points of \(X\) to positive integers with the property that the induced linear map \(\deg \colon \operatorname{Div}(X) \to \mathbb {Z}\) is zero on all principal divisors.

In practice, we can work with a Dedekind domain \(R\) and a function \(\deg \colon \operatorname{Div}X \to \mathbb {Z}\) which is positive on nonzero effective ideals and has the additional property that for any nonzero \(a,b,c \in R\) with \(a+b+c=0\),

By an abstract curve, we mean a connected, separated, regular \(1\)-dimensional Noetherian scheme \(X\). In other words, \(X\) is a connected separated scheme which is covered by open subspaces which are the spectra of Dedekind domains.

For \(F\) a vector bundle on \(X\), a complete flag on \(F\) is a filtration of the form

in which \(F_i/F_{i-1}\) is a vector bundle of rank \(1\) for \(i=1,\dots ,n\). Given a complete flag, define the slope sequence as the tuple \((\deg (F_i/F_{i-1}))_{i=1}^n \in \mathbb {Z}^n\).

By Lemma 35, the function \(\deg \) extends uniquely to a multiplicative function \(\deg \colon \operatorname{Frac}R \to \mathbb {Z}\cup \{ -\infty \} \).

Since the degree map \(\operatorname{Div}(X) \to \mathbb {Z}\) vanishes on principal divisors, it induces a map \(\operatorname{Pic}(X) \to \mathbb {Z}\) which we use to define a degree homomorphism \(\deg \colon K_0(X) \to \mathbb {Z}\).

If \(F \subseteq F'\) is an inclusion of vector bundles of the same rank, then \(\deg (F) \leq \deg (F')\) with equality if and only if \(F' = F\).

We say that \(R\) is factor-filtered if for all \(x,y \in R\),

where \(\deg \) denotes the factorization degree (Definition 34).

Define the factorization degree as the function \(\deg \colon R \to \mathbb {N}\cup \{ -\infty \} \) taking \(0\) to \(-\infty \) and any nonzero \(x \in R\) to the number of factors (counting multiplicity) in the prime factorization of \(x\). (This can be shown to be equal to the length of the ring \(R/(x)\) but we will not use this interpretation.)

A generalized projective line is an abstract complete curve \(X\) satisfying the following conditions.

1. Every divisor of degree \(0\) is principal. In other words the map \(\operatorname{Pic}(X) \to \mathbb {Z}\) is an isomorphism; let \(\mathcal{O}_X(n)\) be the line bundle corresponding to \(n \in \mathbb {Z}\).

2. We have \(H^1(X, \mathcal{O}_X) = 0\).

3. There exists a closed point \(x \in X\) of degree \(1\) with algebraically closed residue field.

In practice, we interpret \(X = \operatorname{Spec}R \cup \{ \infty \} \) in which, for the function \(\deg \colon R \to \{ -\infty \} \cup \mathbb {Z}\) mapping \(r\) to the number of irreducible factors of \(r\) (counting multiplicity),

• For all \(a,b,c \in R\) with \(a+b+c = 0\), \(\deg (c)\leq \max \{ \deg (b) \deg (c)\} \).

• For all \(r,s \in R\) with \(s \neq 0\), there exist \(t,u \in R\) with \(\deg (u) \leq \deg (s)\) and \(r = st+u\).

In what follows, let \(P := \bigoplus _{d \geq 0} P_d\) be a fixed graded ring satisfying the following conditions.

• The subring \(P_0\) of \(P\) is a field and \(P_1 \neq 0\).

• The multiplicative monoid

  \[ \left( \bigcup _{n=0}^\infty P_n \setminus \{ 0\} \right)/P_0^\times \]

  is generated by \((P_1 \setminus \{ 0\} )/P_{0}^{\times }\).

• For each nonzero \(s \in P_1\), there exists a graded ring isomorphism

  \begin{equation} \label{eq:graded ring isom mod s} P/sP \cong P_0 \oplus T L_s[T] \end{equation}

  8

  for some field \(L_s\) containing \(P_0\).

For \(F\) a vector bundle on \(X\), the filtration on \(F\) described in Proposition 28 is called the Harder–Narasimhan filtration, or HN filtration, of \(F\). We define the multiset of HN slopes of \(F\) to consist of \(\mu _i = \deg (F_i/F_{i-1})/\operatorname{rank}(F_i/F_{i-1})\) with multiplicity \(\operatorname{rank}(F_i/F_{i-1})\).

For \(P\) as in Definition 57, for \(d \in \mathbb {Z}\), define the line bundle \(\mathcal{O}_X(d)\) on \(X\) as \(\operatorname{Proj}\bigoplus _{n=0}^\infty P_{n+d}\).

A local field is a complete discretely valued field with finite residue field. Any such field is isomorphic to either a finite extension of \(\mathbb {Q}_p\) for some prime \(p\) or \(\mathbb {F}_q((t)\) for some finite field \(\mathbb {F}_q\).

Let \(F\) be a nonzero vector bundle on \(X\). The bundle \(F\) is polystable if it can be written a direct sum of stable vector bundles, all of the same slope.

A field \(F\) is real if \(-1\) is not a sum of squares in \(F\). A field \(F\) is real closed is all of the following conditions hold:

• \(F\) is real;

• for every \(x \in F\), one of \(x\) or \(-x\) is a square in \(F\);

• every odd-degree polynomial over \(F\) has a root in \(F\).

Let \(F \subseteq G\) be an inclusion of vector bundles on an abstract curve \(X\). Then there is a unique intermediate bundle \(F'\) such that \(\operatorname{rank}(F) = \operatorname{rank}(F')\) and \(G/F'\) is a vector bundle, called the saturation of \(F\) in \(G\).

Let \(F\) be a nonzero vector bundle on \(X\). The bundle \(F\) is semistable if there is no nonzero subbundle \(G\) of \(F\) with \(\mu (G) {\gt} \mu (F)\).

Let \(F\) be a nonzero vector bundle on \(X\). We define the slope of \(F\) as

Let \(F\) be a nonzero vector bundle on \(X\). The bundle \(F\) is stable if there is no nonzero proper subbundle \(G\) of \(F\) with \(\mu (G) \geq \mu (F)\).

Let \(F\) be a vector bundle on \(X\) admitting a complete flag with slope sequence \(-1,0,\dots ,0,1\) (with \(n\) zeroes for some nonnegative integer \(n\)). Then \(H^0(X, F) \neq 0\).

Let \(F\) be a nonzero vector bundle on \(X\) of nonnegative degree. Then \(H^0(X, F) \neq 0\).

Let \(E\) be the field \(H^0(X, \mathcal{O}_X)\) (Corollary 27); then

In particular, \(H^0(X, \mathcal{O}_X(1)) \neq 0\).

Let \(F\) be a field containing a square root of \(-1\) and admitting a finite algebraically closed extension. Then \(F\) is algebraically closed.

Let \(F\) be a field containing a square root of \(-1\) and admitting a finite purely inseparable extension \(K\) which is algebraically closed. Then \(F\) is algebraically closed.

Let \(F\) be a field containing a square root of \(-1\) and admitting a finite separable extension \(K\) which is algebraically closed. Then \(F\) is algebraically closed.

Let \(F\) be a field which admits an algebraically closed extension \(K\) such that \(K/F\) is Galois of prime degree \(p\). Then \(p = 2\).

Let \(E\) be the field \(H^0(X, \mathcal{O}_X)\) (Corollary 27). Then one of the following conditions hold.

1. We have \(\dim _E H^0(X, \mathcal{O}_X(1)) = 2\) and \(X \cong \mathbf{P}^1_E =\operatorname{Proj}E[x,y]\) (the classical case). In this case, \(H^1(X, \mathcal{O}_X(-1)) = 0\).

2. We have \(\dim _E H^0(X, \mathcal{O}_X(1)) = 3\) and \(X \cong \operatorname{Proj}E[x,y,z]/(x^2+y^2+z^2)\) (the twistor case). In this case, there is an isomorphism \(X \times _E E(\sqrt{-1}) \cong \mathbf{P}^1_{E(\sqrt{-1})}\) via which \(\mathcal{O}_X(1)\) corresponds to \(\mathcal{O}_{\mathbf{P}^1}(2)\).

3. We have \(\dim _E H^0(X, \mathcal{O}_X(1)) = \infty \).

Let \(E\) be the field \(H^0(X, \mathcal{O}_X)\) (Corollary 27). Suppose that \(\dim _{E} H^0(X, \mathcal{O}_X(1)) = \infty \). Then for every positive integer \(d\) and every \(c \in \mathbb {Z}\) coprime to \(d\), there exists a stable vector bundle on \(X\) of rank \(d\) and degree \(c\).

Let \(E\) be the field \(H^0(X, \mathcal{O}_X)\) (Corollary 27). For every integer \(d\) with \(1 \leq d {\lt} \dim _{E} H^0(X, \mathcal{O}_X(1))\), there exists a stable vector bundle on \(X\) of rank \(d\) and degree \(1\).

Let \(F\) be a field which admits an algebraically closed extension \(K\) such that \(K/F\) is Galois of prime degree \(p\). Then \(p \neq 0\) in \(F\).

For \(P\) as in Definition 57, we have \(\dim _{P_0} P_1 \geq 2\). In particular, there exist \(s,t \in P_1\) which are linearly independent over \(P_0\).

For every semistable vector bundle \(F\) on \(X\), \(F^\vee \otimes F\) is trivial.

Suppose that \(R\) is factor-filtered. Then \(-\deg \) extends to a valuation on \(\operatorname{Frac}R\).

For \(x \in R\), \(\deg (x) = 0\) if and only if \(x \in R^\times \).

For \(x,y \in R\), \(\deg (xy) = \deg (x) + \deg (y)\).

Every vector bundle on \(X\) admits a complete flag.

Let \(K/F\) be a finite Galois extension of fields. Then for any \(g \in \operatorname{Gal}(K/F)\) and \(x \in K\), if \(g(x) = x\), then \(x\) belongs to the fixed field of the cyclic subgroup of \(\operatorname{Gal}(K/F)\) generated by \(x\).

We have \(H^1(X, \mathcal{O}_X) = 0\).

Let \(F,G\) be two stable vector bundles on \(X\) of the same slope \(\mu \). Then any nonzero morphism \(f\colon F \to G\) is an isomorphism.

Let \(K/F\) be a quadratic extension of fields of characteristic \(\neq 2\). Then there exists \(a \in F\) which is not a square.

Let \(F\) be a field. Then every polynomial over \(F\) of odd degree has an irreducible factor of odd degree.

Let \(F\) be a vector bundle on \(X\) with all HN slopes in \([0,1]\). Then there exists an exact sequence

of vector bundles on \(X\) such that both \(G\) and \(H(-1)\) are trivial.

Let \(F\) be a field. Let \(p\) be an odd prime. Then for \(a \in F\), \(a\) is not a \(p\)-th power if and only if for every positive integer \(n\), the polynomial \(x^{p^n}-a\) is irreducible over \(F\). (Note that this fails for \(p=2\), for example \(x^4+4 = (x^2-2x+2)(x^2+2x+2)\).)

Let \(F\) be a field. Suppose that \(-1\) is not a square in \(F\) and the field \(K := F[x]/(x^2+1)\) is algebraically closed. Then \(F\) is real closed.

Let \(F,G\) be two nonzero vector bundles on \(X\). If \(F \otimes G\) is semistable, then so are \(F\) and \(G\).

Let \(n\) be a positive integer. Let \(m_1,\dots ,m_n\) be a sequence of integers. Suppose that there are no indices \(1 \leq i {\lt} j \leq n\) such that

Then for \(1 \leq i {\lt} j \leq n\),

Let \(n\) be a positive integer. Let \(c\) be an integer. Let \(m_1,\dots ,m_n\) be a sequence of integers satisfying the following conditions.

1. We have \(m_1 + \cdots + m_n = c\).

2. For \(1 \leq i {\lt} j \leq n\), \(m_j \leq m_i+1\).

Then \(m_n \leq \lfloor \frac{c+n-1}{n} \rceil \) (i.e., \(\lceil \frac{c}{n} \rceil \)).

Let \(n\) be a nonnegative integer. Let \(c\) be an integer. Let \(S\) be a subset of \(\mathbb {Z}^n\) satisfying the following conditions.

1. The set \(S\) is nonempty.

2. For every tuple \((m_1,\dots ,m_n) \in S\), we have \(m_1 + \cdots + m_n = c\).

3. For \(i=1,\dots ,n-1\), there exists a constant \(c_i \in \mathbb {Z}\) such that for every tuple \((m_1,\dots ,m_n) \in S\), \(m_1 + \cdots + m_i \leq c_i\).

4. For every tuple \((m_1,\dots ,m_n) \in S\), if there exist indices \(1 \leq i {\lt} j \leq n\) such that

   \begin{equation} \label{eq:sequence condition3} m_k = m_i + 1 \quad (i {\lt} k {\lt} j), \qquad m_j \geq m_i + 2, \end{equation}

   2

   then there exists another tuple \((m'_1,\dots ,m'_n) \in S\) with

   \[ m'_1 + \cdots + m'_i {\gt} m_1 + \cdots + m_i, \qquad m'_1 + \cdots + m'_k \geq m_1 + \cdots + m_k \quad (k \neq i). \]

Then there exists a tuple \((m_1,\dots ,m_n) \in S\) with \(m_n \leq \lfloor \frac{c+n-1}{n} \rceil \).

Let \(F\) be a field. Suppose that there exists a quadratic extension \(K/F\) which is algebraically closed. Then for each \(a,b \in F\), one of \(a^2+b\) or \(-b\) is a square in \(F\).

Let \(F\) be a vector bundle on \(X\) admitting a complete flag with degree sequence \(m_1,\dots ,m_n\). Then for \(i=0,\dots ,n\), the degree of any rank-\(i\) subbundle of \(F\) is at most

Let \(F\) be a field of prime characteristic \(p\). Let \(K/F\) be a Galois extension of degree \(p\). Then there exists \(z \in K\) with minimal polynomial over \(K\) of the form \(x^p-x-a\) for some \(a \in F\).

Every vector bundle \(F\) on \(X\) admits a unique filtration

in which each successive quotient \(F_i/F_{i-1}\) is a semistable vector bundle of some slope \(\mu _i\) and \(\mu _1 {\gt} \cdots {\gt} \mu _l\).

Every vector bundle on \(X\) splits (nonuniquely) as a direct sum of stable bundles. In particular, the HN filtration always splits and every semistable bundle is polystable.

For \(P\) as in Definition 57, \(X := \operatorname{Proj}(P)\) is an abstract complete curve for the degree function carrying every closed point to \(1\).

The degree map \(\deg \colon \operatorname{Div}(X) \to \mathbb {Z}\) induces an isomorphism \(\operatorname{Pic}(X) \cong \mathbb {Z}\).

For \(P\) as in Definition 57, for each nonzero \(s \in P_1\), \(P[s^{-1}]_0\) is a principal ideal domain.

Every semistable bundle of degree \(0\) on \(X\) is trivial.

The tensor product of any two semistable vector bundles on \(X\) is semistable.

Set

Then the following statements hold.

1. There exists exactly one isomorphism class of stable vector bundles \(\mathcal{O}_X(\lambda )\) of slope \(\lambda \) if \(\lambda \in S\) and none if \(\lambda \notin S\).

2. For \(\lambda \in S\) with denominator \(d\) in lower terms, \(\operatorname{rank}(\mathcal{O}_X(\lambda )) = d\).

3. The endomorphisms of \(\mathcal{O}_X(\lambda )\) form a division algebra of degree \(d\) over \(P_0\).

4. For \(\lambda , \lambda ' \in S\), \(\mathcal{O}_X(\lambda ) \otimes \mathcal{O}_X(\lambda ') \cong \mathcal{O}_X(\lambda +\lambda ')^{\oplus m}\) for some positive integer \(m\).

5. Every vector bundle on \(X\) splits (nonuniquely) as a direct sum of summands each of the form \(\mathcal{O}_X(\lambda )\) for some \(\lambda \in S\).

Let \(F\) be a field which admits a finite algebraically closed extension. Then \(F\) is algebraically closed or real closed.

For \(P\) as in Definition 57, \(X := \operatorname{Proj}(P)\) is a generalized projective line. In particular the conclusion of Theorem 56 applies.

Any exact sequence of vector bundles on \(X\) of the form

in which every HN slope of \(F_1\) is greater than or equal to every HN slope of \(F_2\), is split.