On the rank of the flat unitary summand of the Hodge bundle

<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="f colon upper S right-arrow upper B"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:<! -- : --></mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">→<! -- → --></mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f\colon S\to B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a nonisotrivial fibered surface. We prove that the genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="u Subscript f"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">u_f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unitary summand of the Hodge bundle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="f Subscript asterisk Baseline omega Subscript f"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>∗<! -- ∗ --></mml:mo> </mml:msub> <mml:msub> <mml:mi>ω<! -- ω --></mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">f_*\omega _f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the Clifford index <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="c Subscript f"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">c_f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the inequality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="u Subscript f Baseline less-than-or-equal-to g minus c Subscript f"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>≤<! -- ≤ --></mml:mo> <mml:mi>g</mml:mi> <mml:mo>−<! -- − --></mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">u_f \leq g - c_f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, we prove that if the general fiber is a plane curve of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mo>≥<! -- ≥ --></mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\geq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the stronger bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="u Subscript f Baseline less-than-or-equal-to g minus c Subscript f Baseline minus 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>≤<! -- ≤ --></mml:mo> <mml:mi>g</mml:mi> <mml:mo>−<! -- − --></mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>−<! -- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">u_f \leq g - c_f-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> holds. In particular, this provides a strengthening of bounds proved by M. A. Barja, V. González-Alonso, and J. C. Naranjo and by F. F. Favale, J. C. Naranjo, and G. P. Pirola. The strongholds of our arguments are the deformation techniques developed by the first author and by the third author and G. P. Pirola, which display here naturally their power and depth. </p>