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• W4289595188 abstract "For a hyperkähler manifold <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 n> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, Huybrechts showed that there are constants <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a 0> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>a_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a 2> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>a_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript 2 n> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>a_{2n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi left-parenthesis upper L right-parenthesis equals sigma-summation Underscript i equals 0 Overscript n Endscripts StartFraction a Subscript 2 i Baseline Over left-parenthesis 2 i right-parenthesis factorial EndFraction q Subscript upper X Baseline left-parenthesis c 1 left-parenthesis upper L right-parenthesis right-parenthesis Superscript i> <mml:semantics> <mml:mrow> <mml:mi>χ<!-- χ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:mfrac> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>i</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>!</mml:mo> </mml:mrow> </mml:mfrac> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} chi (L) =sum _{i=0}^nfrac {a_{2i}}{(2i)!}q_X(c_1(L))^{i} end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> for any line bundle <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=application/x-tex>L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript upper X> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>q_X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Beauville–Bogomolov–Fujiki quadratic form of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here the polynomial <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation Underscript i equals 0 Overscript n Endscripts StartFraction a Subscript 2 i Baseline Over left-parenthesis 2 i right-parenthesis factorial EndFraction q Superscript i> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:mfrac> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>i</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>!</mml:mo> </mml:mrow> </mml:mfrac> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>sum _{i=0}^nfrac {a_{2i}}{(2i)!}q^{i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is called the Riemann–Roch polynomial of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, we show that all coefficients of the Riemann–Roch polynomial of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are positive. This confirms a conjecture proposed by Cao and the author, which implies Kawamata’s effective non-vanishing conjecture for projective hyperkähler manifolds. It also confirms a question of Riess on strict monotonicity of Riemann–Roch polynomials. In order to estimate the coefficients of the Riemann–Roch polynomial, we produce a Lefschetz-type decomposition of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal t normal d Superscript 1 slash 2 Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>t</mml:mi> <mml:mi mathvariant=normal>d</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathrm {td}^{1/2}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the root of the Todd genus of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, via the Rozansky–Witten theory following the ideas of Hitchin and Sawon, and of Nieper-Wißkirchen." @default.
• W4289595188 created "2022-08-03" @default.
• W4289595188 creator A5065199727 @default.
• W4289595188 date "2022-08-03" @default.
• W4289595188 modified "2023-09-30" @default.
• W4289595188 title "Positivity of Riemann–Roch polynomials and Todd classes of hyperkähler manifolds" @default.
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• W4289595188 doi "https://doi.org/10.1090/jag/798" @default.
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