Riemann-Roch theorem for normal surfaces and applications
In the theory of smooth compact complex surfaces, the Riemann-Roch formula \((\chi({\mathcal O}_X(D))= \chi(X)+\frac12 D(D- K_X))\) and the adjunction formula \((2p_a(C)-2= C^2+CK_X)\) are two of the most important foundational results. The purpose of this article is to work out these formulas in the case of normal projective surfaces and to show how they fit into various applications. Special emphasis is given to the case of projective orbisurfaces (i.e. surfaces with at most quotient singularities) since the results are much better in this more restricted setting. The basic problem which occurs in passing from smooth surfaces to normal surfaces is that there are, in general, Weil divisors on the surface which are not Cartier divisors. As a consequence, the intersection pairing has values in \(\mathbb{Q}\) instead of \(\mathbb{Z}\), and the RR (= Riemann Roch) formula and the adjunction formula need to be adjusted by correction terms. These correction terms can be localized at the finitely many singular points of \(X\). We now give a brief outline of the results in this article. The generalized RR theorem for Weil divisors on normal surfaces is given in (1.2). The correction terms (which turn out to be maps \(R_{X,x}: \text{Weil/Cart} (X,x)\to \mathbb{Q}\), where \(x\in \text{Sing }X\)) are calculated explicitly; they consist of an intersection theoretical contribution and a cohomological contribution. In (2.1) we state the generalized adjunction formula. The correction terms possess an additional symmetry, which stems from Serre duality. We use the previous mentioned results in order to give quick proofs of two results by Zariski: On a normal surface, the arithmetic genus \(p_a\) of curves can be naturally generalized to a map \(p_a: \text{Weil } X\to\mathbb{Z}\), and \(\chi({\mathcal O}_X(D))= \chi(X)-1+p_a(-D)\) for all \(D\in\text{Weil } X\). -- \S 4 contains two applications to embedded curves: Here again local correction terms appear, which are entirely expressed in terms of the principal correction terms \(R_{X,x}\). It is conjectured that there is a universal constant \(N\) such that \(P_N(X)\geq 2\) for every minimal orbisurface of general type. We show that \(P_n(X)\geq 2\) implies that \(K_X^2\geq \frac{1}{2n^3+n^2}\) for every such surface \(X\). As an application of our explicit calculation in (5.2) of the correction terms \(R_{X,x}(nK_X)\), we can prove that \(P_2(X)\geq 2\) (and hence \(K_X^2\geq \frac{1}{20}\)) for every minimal orbisurface \(X\) of general type, in case \(E_i^2\leq-4\) for all the irreducible components \(E_i\) of the exceptional curve \(E\) of the minimal resolution of singularities of \(X\). In \S 6, we give a new construction of the second Chern class of a reflexive sheaf of arbitrary rank on a projective orbisurface. The generalized RR theorem (1.2) is further generalized to arbitrary reflexive sheaves in (6.3). The result of our construction agrees with the result of two different approaches by Kawamata, cf. (6.12). -- In \S 7, we apply the constructions of \S 6 to the concrete sheaf \(\Omega_X^1\) of holomorphic 1-forms on \(X\). We calculate \(R_{X,x} (\Omega_X^1)\) (theorem 7.2) and \(c_2(\Omega_X^1)\) (theorem 7.3). -- The explicit formula (5.2) for \(R_{X,x}(nK_X)\) is applied in \S 8 to give an explicit formula for the local plurigenera of a quotient germ, cf. (8.1). We briefly discuss the relation between local and global plurigenera in (8.2). In (8.4) we give one more concrete example for which we can calculate both the correction terms \(R_{X,x}(nK_X)\) and the local plurigenera \(P_n(X,x)\) explicitly. Our proofs often rely on certain globalizations of local data; hence we have to provide ourselves with globalization theorems (which are also interesting in their own right).
scientific article; zbMATH DE number 878389
scientific article